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INTRODUCTORY SCIENCE TEXTS 



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Noise 
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QUANTUM ELECTRONIC 

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W.P. JOLLY 



LOW NOISE 
ELECTRONICS 



This book is one of a series of short, 
simple, treatments of advanced modern 
subjects. It has been written 
so that the whole range of the topic 
may be grasped quickly by a reader 
with the general scientific background 
of a first or second year college 
student. The text is supported by 
references to books and journals 
carefully chosen to extend any section 
well into the post-graduate field. 

The book should be valuable to 
working graduates who wish to keep 
up to date, particularly science 
teachers, to students of electrical 
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Low noise Electronic 



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LOW NOISE ELECTRONICS 

An introduction to quantum and electron beam electronics 



Introductory Science Texts 

Editor: Professor W. P. Jolly 



LOW NOISE ELECTRONICS 

W. P. Jolly. B.Sc, C.Eng., F.I.E.E. 

Assistant Professor in the Dept. of Physics and Electrical Engineering 
Royal Navat College, Greenwich, 

HIGH FREQUENCY COMMUNICATIONS 

J, A. Bctts, B.Sc, Ph D., C.Eng., M.I.E.E. 

Department of Electronics 
University of Southampton, 



Low Noise 
I Electronics 



In preparation: 

HYBRID COMPUTATION 
CRYOELECTRONICS 



W. P. JOLLY, B.Sc, C.Eng., F.I.E.E. 

Assistant Professor in the Department of Physics 
and Electrical Engineering 
Royal Naval College, Greenwich 



!k> isa 




THE ENGLISH UNIVERSITY 

St Paul's House Warwick <(ahe London 



First printed 1967 



Copyright © 1967 
W. P. Jolly 



Printed and bound in Great Britain for 

THE ENGLISH UNIVERSITIES PRESS LIMITED 

by Richard Clay (The Chaucer Press), Ltd., 
Bungay, Suffolk 



Editor's Foreword 



The very rapid growth of engineering and the physical sciences in 
recent years has meant that many new subjects have appeared which 
are so important fundamentally and technologically that they could 
well be studied in an undergraduate course. The degree and diploma 
syllabuses are, however, already so crowded that little time can be found 
for new subjects and a post-graduate year of extra tuition is generally 
proposed to meet this difficulty. 

There are many students and busy practising scientists and engineers 
who will wish to have a working knowledge of important and growing 
new subjects but who have no time to study them at length. 

The books in this series are intended to be short and simply written 
so that they may be read in a few days by anyone with the general back- 
ground of a second-year student. 

The text is supported by references to published papers and textbooks, 
chosen from the large mass of available material as being most likely 
to help the reader who may wish to pursue the subject, or some part of 
it, in greater depth now or later. 

W. P. Jolly 



Author's Preface 



I have tried to produce a book which really is an introduction to the 
subject. Masers, lasers, electron beam tubes, parametric amplifiers and 
associated topics in low noise and quantum electronics have been treated 
simply, and any supporting material needed, e.g. semiconduction, has 
been developed from fundamentals. 

Sometimes the full and detailed discussion of a principle has been 
thought too long or difficult for a book which aims to be a primer. In 
such cases the principle is stated and discussed simply in the text, and 
reference made to an original paper, review article, or advanced textbook 
where a full account may be found. 

The bibliography lists a number of books for the reader who wishes 
to extend his knowledge, and suggests journals which should be con- 
sulted from time to time to keep informed of new developments. 

There are no toy cricket bats at Lords but many children get the feel 
of the game by using a bat light enough to handle with ease. I would 
like to feel that this book filled a similar role. 

W. P. Jolly 



Contents 

Editor's Foreword v 

Author's Preface vii 

1. Continuous Spectrum Radiation 1 

2. Radiation at Discrete Frequencies 11 

3. Quantum Picture of Electrons in Solids 17 

4. Interaction of Photons, Phonons, and Electrons 31 

5. Radiation and Matter — Einstein's Radiation Theory 41 

6. Non- Equilibrium Systems 50 

7. Power Relations in Parametric and Non- Equili- 

brium Systems 60 

8. Parametric Diode and Ferrite Amplifiers 70 

9. Parametric Electron Beam Amplifiers 81 

App. I Practical Maser Systems 90 

App. tl Laser Systems and Applications 98 

App. Ill Electron Beam Amplifiers 108 

App. IV Ferrites and Their Microwave Applica- 
tions 1 22 

App. V Injection Laser 1 29 

App. VI Requirements for Low Noise Amplifi- 
cation Near 1 Gc/s 135 

Bibliography 141 

Index 143 






1 

Continuous Spectrum Radiation 



sources of electromagnetic radiation usually generate energy either 
over a wide continuous band of frequencies, or at a comparatively small 
number of discrete frequencies. Typically, a hot solid is a source of the 
first kind and an electrically excited gas in a discharge tube is a source 
of the second kind. 

The study of the continuous spectrum of radiation from a hot solid 
led Planck, in 1900, to formulate the quantum theory. This chapter will 
be concerned with the quantum theory of radiation and its special 
application in modern communications. The next chapter will show 
how Bohr applied the quantum theory to explain the line spectra of the 
radiation emitted by sources of the second kind. 

Let us first review some of the ideas about continuous spectrum 
radiation which were already established before the quantum theory. 

Pre-quantum Ideas 

By the end of the nineteenth century a number of important qualitative 
concepts about radiation had become accepted. 

It was known that good absorbers were good radiators, and that the 
best radiator was therefore a black body. This was defined as a body 
which absorbed completely all the radiation failing on it. Also the 
theory of exchanges had shown that the transfer of energy by radiation 
from a hot to a cold body was a nett process — a hot body radiating 
energy to a cold one was at the same time receiving energy from it. 

A constant temperature enclosure containing various bodies will be 
filled with radiation as energy passes to and fro between the bodies and 
the walls. Each body radiates energy of exactly the same amount and 
character as it absorbs, otherwise it would change its temperature. 
Thus the energy density of the radiation in the enclosure, and its 
spectral distribution, depend only on the temperature. 

A black body in the enclosure will absorb all the energy falling on it 
and will thus radiate the same energy. Thus the amount and spectral 



2 LOW NOISE ELECTRONICS 

character of the radiation emitted by a black body is exactly the same as 
that which falls on the body in an enclosure at the same temperature. 
In addition to qualitative ideas of this type there were two important 
pre-quantum laws of a quantitative kind. These are Kirchhoff's and 
Stefan's laws which are dealt with below. 

Kirchhoff's Law 

Kirchhoff's law states the connection between ex . dx, the emissive power, 
and the absorptive power of a body at a given temperature. 

If a body emits ex . dx of energy per unit area per second in the form 
of radiation with wavelength lying between X and X + dx, then e A is called 
the emissive power. 

If a given amount of radiation in this same waveband falls on a body 
and a fraction ax is absorbed, then ax is called the absorptive power. 

Consider unit area of any body in a constant temperature enclosure. 
The radiation falling on it is dQ r and the amount absorbed ax . dQ 
equals the radiation emitted by it and depends only on the temperature. 



Thus 



ex.d\ = ax.dQ 

eh^dQ 

ax dx 



which is constant for all bodies at a given temperature. 

Stefan's Law 

Stefan's law is concerned with the total power e of all wavelengths 
emitted by unit area of a black body at temperature T. 

e = oT 4 

where a is Stefan's constant 5-735 x 10~ 8 watt/m^deg 4 . 

With Stefan's law established for the total amount of radiation of all 
wavelengths emitted by a black body it only remained to show how this 
energy was distributed among the various wavelengths. 

Distribution Laws. Planck's Formula 

If a satisfactory explanation were to be provided for the spectrum of 
radiation contained in a constant temperature enclosure, or emitted by 
a black body, then a plausible statement had first to be made about the 
number and behaviour of the individual sources of energy at the various 
frequencies. Statistical mechanics and classical wave theory provided 
such a statement. 












CONTINUOUS SPECTRUM RADIATION 3 

Statistical mechanics is concerned with the use of the laws of 
mechanics and of probability to predict the equilibrium state of a 
system. Such a treatment shows that, of a large number of oscillators 
of a given frequency, the number possessing energy between E and 

B_ 

E + dE is equal to Ae~*r . dE, where A is a constant depending on 
the total number of oscillators present, T is the absolute temperature, 
and k is Boltzmann's constant. This is Boltzmann's law and is fully 
treated in Jean's Dynamic Theory of Gases. 

Planck introduced his novel quantum concept and used it in the 
Boltzmann expression to obtain the mean energy of all the oscillations 
in the frequency range/ to /-f- df. He suggested that any oscillator at 
frequency /could only lose or gain energy in multiples of hf the single 
quantum. Thus the values which E can take in the Boltzmann ex- 
pression are 0, hf 2hf 3hf etc. 

The number of oscillations with these energies will be N, Ne~ h f tkT , 
tf e s*ff*T i etc ^ w here N is the number present in the lowest energy 
state. 

Summing all these terms gives the total number of oscillations present 
as 

w . (a) 

1 - e« 

Total energy present is N{hferMkT + 2hfe~ 2h f! kT + , . .) 



Nhf 



A/v2 



e*r \1 — e *t) 



m 



Classical wave theory may now be used to calculate the number of 
oscillations of frequency between/ and/ -f- df in unit volume of an 
enclosure. This figure is then equated to {a) above, so that the unknown 
Nmay be eliminated from (b) which gives the total energy at frequency/. 

The number of oscillations per unit volume is found to be -zydX or 
87^2 



~§- df where c is velocity of light. 1 



Thus 



N _ 83P , f 
1 _ e-htikT - c s a J 



N = (l 



e-v/*T) ^ d f 



Roberts, Heat and Thermodynamics (1940), p. 416. 



LOW NOISE ELECTRONICS 



and substituting in (b) gives for the total energy per unit volume in the 
frequency range/ to / + df 



E f .df = 



Uff_ 

C Z e hfikT (J _ e ~hflkT\ 



¥ 



df 



= 8*/*/ hf_ 



[ehfikT^ i) d f 



which is Planck's distribution law. 
If Ef . dfis the energy density, then the energy falling per second on 

c 

unit area in the enclosure is Ef . df x .. and this is the power radiated 

by unit area of a black body. 2 

Note that dividing (b) by (a) gives for the mean energy of each 
oscillation 

\ehftkT- \j 
instead of the classical equipartition value kT. 

Comparison of Quantum and Classical Views of Radiation 

The difference between Planck's law and earlier, classical distribution 
laws is the substitution for kT of the frequency dependent expression 

¥ 



e hflkT _ J 

as the energy associated with an oscillator. 

e hflkT may be expanded as 1 + &, + M -g=\ and if hf < kT, then the 

quantum expression reduces to the classical value kT and Planck's law 
reduces to the Rayleigh-Jeans expression 

Sir/" 2 

E f .df=^kT.df 

The classical law fails if the frequency is high (the ultra-violet 
catastrophe), or if the temperature is low. The more accurate quantum 
expression must then be used. 

Electronic Noise 

The infinite frequency spectrum of radiation from a black body in- 
cludes a significant power output in the "radio" frequency band. This 
radiation will be detected by a radio receiver and its presence will 
3 Roberts, Heal and Thermodynamics (1940), p. 388. 



CONTINUOUS SPECTRUM RADIATION 5 

indicate that the black body is delivering power to the input terminals 
of the receiver either through the aerial or in some other fashion. This 
power will be amplified in the receiver and will appear at the output 
terminals, where it will in general act as a background against which 
the wanted signal must be detected. 

The ability of the receiver to detect a signal is determined by the 
signal to noise power ratio, and the noise may arise from thermal 
radiators outside the receiver or from circuit elements within it. In 
particular, a resistance across the input terminals of a receiver will act 
as a noise source. 

Nyquist 3 in 1928 considered the particular case of a resistance as a 
noise source and derived the expression kTB for the available noise 
power in the bandwidth B. The Nyquist expression, which is of con- 
siderable generality, was originally derived in classical equipartition 
terms and the term kT should more properly be replaced by the quan- 
tum term. 

But just as Planck's accurate quantum distribution law reduces to the 
classical Rayleigh-Jeans expression when hf 4, kT, so the kTB ex- 
pression is of sufficient accuracy in many radio fields. 

It is instructive to compare the "classical" and the "correct" noise 
powers in the microwave region, at room temperature and at a typical 
cryogenic temperature. 

For ease of calculation let us take 30,000 Mc/s, i.e. X = 1 cm, as a 
typical microwave frequency. 

We are concerned with the discrepancy between kT and 

¥__ 

e hflkT _ i 



i.e. between M and , j. 1 1 + | -r4 J if only the first two terms of the 
expansion are considered. 

The comparison between 1 and [ 1 + —= J gives an indication of the 
discrepancy. 

For30,000Mc/sat300"K V. = |£ g»x£»X«£ <3 x 10- 

For 30,000 Mc/s at!* K *JL - 0-69 

Thus the use of the classical expression kTB to calculate noise power 
in the microwave band at room temperature leads to little error. 

3 Nyquist, Phys. Rev., Vol. 32, 1928, p. 110. 



LOW NOISE ELECTRONICS 



But at I ° K the calculated noise power using kTB may be very much 
greater (= 50% greater) than that actually experienced. 

Noise Input to Directional Aerial from Large Distant Source 

As an example of the generality of Nyquist let us consider an aerial 
looking out into space. 

NOISE SOURCE 

T 




Fig. 1.1 

The directional aerial has dimensions comparable with the wave- 
length of the radiation being received, and it is in a narrow band 
centred on this wavelength that the noise input is of interest. 

Planck's distribution law for £> . df, the energy density in an en- 
closure at frequency/, is 

Power radiated by unit area of black body in this band is 

£f£ r1f- 2K ( h f W 

4 ' QJ X 2 [ e hmT Z i fJ ( S ee p. 4) 

• x 2 

Diffraction determines the beam width of the aerial as -r> where A is 

A 

area of aerial. 4 
Beam width and range R determine the area of the noise source which 

J? 2 x 2 
radiates to aerial as —. — 
A 

Power radiated towards aerial is 
R*& 2n( 



P = 



III 2 W hf \ Jr 



4 Physics for Electrical Engineers, Jolly, E.U.P. (1961). 



CONTINUOUS SPECTRUM RADIATION 
Power per unit area at aerial is 



Thus power received by aerial is 

pa m z 



2kR z 



X 2 '2TtR*\eW kT - l) J 
- ( ¥ - W- (—K—\n 

~ \ e hflkT _ 1 p — \ e hf\kT _ l) D 



if bandwidth B is small. With the usual reservations this reduces to 
kTB. 

Noise Temperature 

Commonly a radiator at temperature Twill not be a black body but will 
radiate energy as though it were a black body at a different temperature 
T. V is called the colour temperature of the body in pyrometry and the 
noise temperature in electronics. 

Receiver Noise, Noise Input Temperature 

When a signal is amplified and processed by a receiver the ratio of 
signal to noise power at the output is less than that at the input because 
of the extra noise introduced by the various stages inside the receiver. 
The noise figure of a receiver is defined as 



_S i /So 
Nil N 



where S and TV are signal and noise power, and the subscripts refer to 
input and output. 

Each stage of an amplifier will introduce some noise and the effect of 
this noise is greater if it is introduced by an early stage because it will 
then be amplified by the following stages. 

It is convenient to express the effect of all the noise introduced by the 
various stages of the receiver in terms of the noise power Nr^ which 
would have produced the same effect if introduced at the input terminals. 

The power appearing at the output terminals due to noise introduced 
by the receiver is thus GNr^ where G is the power gain of the receiver. 



Thus F = %&=^ = 
Nif No GNi 



GNi + GN Rl N t + N Ri 



GN t 



Nt 



i.e. 



NRi 

N t 



"« = f _ i 



8 



LOW NOISE ELECTRONICS 



Note that it is Nn t which is the fixed receiver property and that the 
noise figure F depends here upon Ni. 

So that F shall also be a fixed receiver parameter, Ni is standardised 
as the noise from a source at 290° K. 

Another way of considering the noise introduced by a receiver is to 
use the concept of receiver noise input temperature. 

The input noise temperature 7V of a receiver is the increase in source 
temperature needed to doubte the output of the receiver when originally 
connected to a noise-free source. 

Thus N Rt = kTiB 
and AT, = 290 . kB 

Whence T ( = (F - 1) x 290° K 

A good conventional microwave receiver might have a noise figure of 
10, giving it a noise input temperature of just over 2600° K. 

Sometimes the noise figure is quoted in decibels, in which case 
T t = 290 (10* /10 - 1), where x is the noise factor in db. 

The new unconventional amplifiers, e.g. masers, generate so little 
noise that their noise input temperature may be only a few degrees. 
In this field it is usual to work in noise temperature, but if a noise figure 
is quoted it will be very close to unity, e.g. F = 1-01 gives 2-9° as input 
temperature. 

The reader may possibly encounter the statement that a noise figure 
of 1*5 corresponds to a noise temperature of 2°. Such a statement will 
refer to some low temperature device operating in liquid helium at 
4° K. This ambient temperature is used instead of the 290° of the 
formal definition. Such an alternative standard is confusing and should 
be avoided. 

Noise Due to Lossy Medium 

If *?a is the emissive power of a real body and &e A is that of a black body 
at the same temperature T, then using the classical approximation. 

Power radiated by black body is be*.d\ = kTB 
Power radiated by real body is e k ,d\ = kTB 
Thus r/r = eV&e A 

But ex may be found in terms of ax the absorption coefficient at this 
temperature. 



CONTINUOUS SPECTRUM RADIATION 



9 






By Kirchhoff's law exjax = bexibax — tfx, where &a A = 1 for black 
body. 



Thus 
and 



e^ — ax. b£\ 
T(T = ax. WbCA 
r = axT 



Any absorbing substance will act as a noise source with a noise 
temperature depending on its absorption coefficient and its real tem- 
perature. The lossy medium may be the atmosphere or an attenuating 
element in the feeder system, such as a lossy switch or waveguide. 

A waveguide at room temperature giving 1% absorption would con- 
tribute about 3° K to the total noise temperature, which might be about 
20° K in a typical low noise system. If the guide were cooled in liquid 
helium the contribution would be negligible (0-04° K). 

Noise Temperature for Complete System 

Published figures 5 for the noise performance of a high-quality system 
are worth studying in terms of some of the ideas developed earlier in 
this chapter. 

Sky noise at zenith . . 2*5° K 
Subsidiary lobes . . . 2° K 

Aerial, joint, and feeder loss 12° K 
Maser noise . . 2° K 



18*5° K 



The sky noise is quoted with the aerial pointing vertically upwards. 
This gives the minimum path through the absorbing atmosphere. At 
lower angles of elevation the path through the lossy atmosphere, which 
is at a relatively high temperature, will be longer and the noise tempera- 
ture consequently higher. 8 

Any high-gain aerial may have small side or back lobes as well as the 
main beam. These may receive radiation from the ground and give 
subsidiary lobe contribution to the total noise temperature. 

The feeders, switches, etc., are treated just as any lossy medium and 

5 "Overall System Requirements for Low Noise Performance", Ditchfield, /. Brit. 
I.R.E., Vol. 22, August 196 J. 
fl See also Sky Noise, p. 1 36. 



10 



LOW NOISE ELECTRONICS 



will give noise depending on the attenuation they introduce and their 
actual temperature. 

The contribution of the maser itself to the total noise temperature of 
the system 7 is very small compared with other types of amplifier and 
is discussed elsewhere. 

7 "System Noise Temperature of Quantum Amplifiers", Van der Ziel, Proc. 
I.E.E.E., June 1963. 






Radiation at Discrete Frequencies 



the nature of the continuous spectrum of electromagnetic radiation 
emitted by black bodies, and by solids which behave in a similar 
fashion, has already been considered, and Planck's quantum theory has 
emerged as a result of the attempts to explain the shape of the energy 
distribution curve. 

In this section will be discussed those substances, principally gases, 
which only emit — or absorb) — radiation of certain discrete wavelengths 
or within certain fairly narrow band widths. The subject stems from 
Bohr's use of Planck's quantum ideas to explain the line spectrum of 
hydrogen in terms of atomic energy levels. 

Bohr Theory of Hydrogen Spectrum 

The hydrogen atom contains only one electron and Bohr postulated 
that the electron rotated round the nucleus in a circular orbit. 

Let the radius of the orbit be r and the electron velocity, mass, and 
charge v f m, and e. 

Then, equating the centrifugal force and the Coulomb force of 
attraction 

mv 2 e 2 



tekor 2 



Bohr further assumed that the angular momentum of the electron 
mvr could only be an integral multiple of y- > where h was Planck's 

constant. 

nh 
mvr = =- 

where n is any integer. 

Eliminating v from these two equations gives 

= rfkoh 2 
v:e 2 fn 



12 



LOW NOISE ELECTRONICS 



The total energy of the system is the kinetic energy of the moving 
electron plus the potential energy due to the separated positive and 
negative charges. 

The potential energy of the electron at radius r is the potential energy 
at infinity £"«, less the work performed as the electron comes in from 
infinity to r. 



Potential energy is E a 



-f- 
JL4* 



kor 



dr = £«, — 



4nkor 



Kinetic energy is \mv z = s— ? — 

oTT/cor 

e 2 rfiknh 2 

Thus total energy is £«, — «— . — and r = — r , — 

Qnkor Tie l m 

If the electron falls from an orbit of large radius r 2 , characterised by 
quantum number «2, to an orbit of smaller radius n characterised by «i, 
then energy E 2 — E\ — hf is radiated. 

hf~(F ^ m \ (f ^ m \ - eim ( l -L\ 

V \** M WJ {*"* 8ko 2 niW}~&k W[ni2 n#) 
Since c =f\ where c is the velocity of light, 



%ktfh*c 



\m 2 m 2 / 



This expression, now deduced theoretically by Bohr, was of exactly 
the same form as that which had been formulated earlier by experi- 
mental spectroscopists to describe the wavelength of the lines of the 
hydrogen spectrum. 

More Complex Atomic Models 

The simple line spectrum of hydrogen can be explained in terms of 
Bohr's model. This model is based on classical mechanics and electro- 
statics with two innovations: the introduction of the Planck relation 
between energy and frequency E = hf, and the quantisation of angular 
momentum. 

There remained a number of other spectral features, well established 
empirically but not explained by Bohr's simple model. Such features 
were fine structure, hyperfine structure, and electro- or magneto-optical 
effects, such as the Stark and Zeeman effects, where the application of 
an electric or magnetic field to the light beam before dispersion in the 
spectrometer produces splitting of the spectral lines. 






RADIATION AT DISCRETE FREQUENCIES 



13 



In order to account for these more detailed spectral properties the 
simple model was elaborated by the addition of such concepts as 
elliptical orbits, electron spin, and nuclear spin. The extra properties 
introduced were quantised, just as angular momentum had been in the 
simple model. As a result the widely separated single electronic energy 
levels of the original model were replaced by sets of closely spaced levels 
which accounted for the fine and hyperfine structure of the spectral 
lines. 

Because of Planck's relation E = hf between energy and frequency 
in a quantum process, it will be necessary for a transition to occur 
between a pair of very closely spaced levels if an atom is to radiate or 
absorb energy in the microwave region. 

Such a transition occurs in the hydrogen atom at 1420 Mc/s, and has 
been detected in interstellar radiation. It is associated with the magnetic 
interaction of nuclear and electron spin, and the energy change involved 
when the electron changes its spin from one direction to the other. 

The caesium atom has a single electron in its outermost orbit. The 
spin of this electron reacts with the nuclear spin in a similar fashion to 
that of the hydrogen atom to produce a transition, in the case of 
caesium, at 9192 Mc/s. This transition is used to produce the frequency 
standard in the caesium atomic clock. The frequency standard is an 
oscillator which supplies energy to the stream of caesium atoms. 
When the oscillator is at precisely the transition frequency a maximum 
amount of power is absorbed. A control system monitors the amount 
of power absorbed and alters the oscillator frequency until maximum 
absorption is obtained. Frequency stability better than one part in 
10 9 has been obtained. 

Molecular Structure 

The energy-level picture for a molecule will be correspondingly more 
complex than that for an atom, because there will be at least two atomic 
systems in close proximity. These will each react upon the other to 
produce increased complexity in the original system. Further energy 
levels are introduced because the molecule may possess vibrational or 
rotational energy. 

So complex is the energy-level structure that the molecular spectra 
tend to consist of bands of extremely closely spaced levels. Microwave 
absorption spectroscopy is a useful way of determining the detailed 
energy-level structure. 

Although it is often possible to discuss some particular feature of a 
molecular spectrum in terms of a model, it is only a quantum mecha- 
nical solution of the problem which reveals the full details of the energy 



14 



LOW NOISE ELECTRONICS 



levels and justification of the selection rules which state whether transi- 
tions between certain levels are permitted or not. 

The ammonia molecule was the first in which microwave effects were 
observed and its study has consistently yielded results important in the 
microwave field. Its inversion resonance at 24 Gc/s was the basis of the 
first maser and can be explained in model terms because there are two 
stable structures for the NHs molecule. One structure has the nitrogen 
atom above the plane of the three hydrogen atoms, and the other has 
the nitrogen atom below the plane. A transition between the two states 
involves a quantum of energy at 24 Gc/s. 

Oxygen and Water Molecules 

Oxygen and water molecules are of particular interest because they 
occur naturally in the Earth's atmosphere, and because their complex 
molecular spectra contain closely spaced energy levels between which 
transitions may occur with the absorption of energy in the microwave 
frequency region. 

Electrically non-polar molecules like oxygen and nitrogen normally 
only have resonances at ultra-violet frequencies, corresponding to 
transitions between two electronic states. But oxygen is paramagnetic 1 
and, because of the permanent magnetic moment of the oxygen mole- 
cule, transitions are permitted by the selection rules between closely 
spaced components of the ground state which give rise to microwave 
resonances. 

The microwave spectrum of the oxygen molecule consists of an 
isolated line at 2-5 mm (120 Gc/s) and a set of about twenty-five major 
lines at around 5 mm (60 Gc/s) which are so close together that they are 
unresolved at atmospheric pressure (see below). 

The water molecule 2 has many pairs of nearly coincident energy 
levels which are close enough together for the absorption to lie in the 
microwave spectrum. In practically all cases, however, the selection 
rules forbid transitions between these levels, and there is only one 
strong absorption line in the microwave region, at 1-35 cm (22 Gc/s). 
The remainder of the permitted transitions give resonance peaks in the 
millimetre and in the infra-red regions. 

Atmospheric Absorption 

The general character of the variation with frequency of atmospheric 
absorption in the microwave region is shown in Fig. 2.1. A more 

1 "Absorption of Microwaves by Oxygen", Van VIeck, Phys. Rev., Vol. 71, No. 7, 
p. 413. 

2 "Absorption of Microwaves by Uneondensed Water Vapour", Van VIeck, ibid 
p. 425. 






RADIATION AT DISCRETE FREQUENCIES 



15 



quantitative graph showing the contribution of atmospheric absorption 
to the total noise temperature of the sky is given elsewhere (p. 137). 
The purpose of this section is merely to indicate why the atmospheric 
absorption spectrum is continuous — although peaky — from 1 Gc/s 
upwards, in spite of the fact that the microwave spectra of oxygen and 
water vapour consist of discrete lines at 22 Gc/s and certain other 
higher frequencies. 



ABSORPTION 




10 10 100 

FREQUENCY IN Gc/5 

Fro. 2.1 

There are three types of mechanism which contribute to the fre- 
quency broadening of a single spectral line. 3 The three types, together 
with the order of magnitude of the broadening produced in a micro- 
wave resonance, are: natural width (<1 cycle), Doppler effect (50 kc/s), 
and collision broadening (several Gc/s at atmospheric pressure). 

The natural width of a line increases with the spontaneous emission 
probability, rather as the Q of an oscillating circuit is lowered by 
loading. The width increases with frequency and is significant at X-ray 
frequencies but negligible for microwaves. 

The Doppler broadening is due to the relative motion of the molecule 
and the observer. It depends on the kind of molecule and its tempera- 
ture, but is typically 50 kc/s, which is not important in the communica- 
tions field, although it may be important in microwave frequency 
standards like the caesium clock. 

The discrete frequencies in the absorption and emission spectrum of a 
given gas are calculated for a single isolated molecule. If another 
molecule were close enough to interact with the first, then the spectrum 
would be modified. Collision broadening is the summation of the fre- 
quency broadening effects caused by interaction of molecules which are 

3 Microwave Spectroscopy, Gordy et a!., Chapman & Hall. 



16 



LOW NOISE ELECTRONICS 






instantaneously close together due to random motion in a natural 
sample of gas. 

The magnitude of the collision-broadening effect depends, among 
other things, upon the nature of the colliding gas molecules and upon 
the pressure. It is, however, large compared with the other two effects 
in the microwave region, and causes the individual oxygen and water- 
vapour lines to run into each other in the microwave spectrum as shown 
in Fig. 2.1. A significant part of the absorption curve shown is due to 
transitions of high absorption in the water molecule which occur in the 
millimetre and infra-red regions beyond 120 Gc/s. Some of these are 
pressure broadened so that the skirts of the lines contribute to the 
absorption at much lower frequencies. 

The theoretical calculation of the strength and shape of an absorption 
line in a given molecule and the experimental results 4 show good 
agreement. The extrapolation of the theory to explain the absorption 
by the atmosphere, with all the uncertainties of its composition, is 
difficult. Experiments produce unexpected results, such as the sug- 
gestion 5 that absorption is much higher than would be expected when 
oxygen and water-vapour lines fall close together. 

Other Line Spectra 

The particular gases discussed above do not make up an exhaustive list 
of those with interesting and important microwave properties. Al- 
though they are the most important in the communication field at 
present (see also, p. 137), there is no reason why other molecules should 
not be important in the future. For instance, recent reports 6 describe 
the detection of the absorption in interstellar space of radiation at 
1660 Mc/s. This is one of the lines in the microwave spectrum of 
hydroxyl (OH) which is thus assumed to exist in space. 

In discussing pressure broadening above we have touched upon the 
effects of interaction between closely spaced molecules. These effects 
are accentuated in the solid state (p. 21), although isolated impurity 
atoms in an inactive host material (e.g. Cr in AI2O3) may still give well- 
defined microwave lines of the type described above. 

4 Schulze and Tolbert, Nature, Vol. 200, November 1963, p. 747. 

6 Tolbert and Straiton, PJ.R.E., March 1961, p. 649. 

a "Radio Observations of OH in the Interstellar Medium", Weinreb et al, 
Nature, Vol. 200, No. 4909, November 1963, p. 829, and ibid., Vol. 208, No. 5009 
October 1965, p. 640. 



Quantum Picture of Electrons in Solids 



when many atoms are packed close together in the solid form most of 
the interesting electrical properties, e.g. conductivity and the way in 
which the material reacts to electromagnetic radiation at optical and 
microwave frequencies, can be explained in terms of the behaviour of 
certain of the electrons in the material. In particular, the energy 
possessed by the electrons and the way in which it may change is 
important. 



Fig. 3.1 

The total energy possessed by the electrons in a solid element is 
distributed among them in a complex fashion, but the elaborate elec- 
tronic energy distribution in the solid is evolved from the more simple 
energy distribution of each constituent atom — the characteristic energy- 
level structure of an isolated atom of the particular element. 

We have already developed a quantitative theory for the electron 
energy levels in the isolated hydrogen atom. It is instructive to consider 
tlie extent to which the general principles established about the energy- 
level structure of hydrogen and other multi-electron atoms can be 
extended into the solid state. 

The permitted energy levels for the electron in the hydrogen atom 
have the general form shown in Fig. 3.1 E\ is the ground or normal 
state for the atom and Et is the maximum energy the electron may 



18 



LOW NOISE ELECTRONICS 



1 



possess without ionisation occurring. The full line for £1 indicates that 
the particular atom shown is in the ground state, i.e. that this energy 
level is occupied by an electron. The dotted lines for the higher energy 
levels indicate that in this case these permitted levels are vacant. 

If the electron possesses more energy than Et, then it is no longer 
confined to one of the permitted orbits near the nucleus but may wander 
farther away without restriction. 

When models of the atoms containing more than one electron were 
constructed and explanations produced for the spectra of these heavier 
elements, another important principle was recognised which will be 



£4 

.E 3 



Fio. 3.2 



F10. 3.3 



extended into the more complex system of the multi-atom solid crystal". 
This principle was Pauli's exclusion principle, which stated that no more 
than two electrons in an atom could possess the same energy, and these 
two electrons must have opposite spins. 

Thus the helium atom in the ground state has two electrons of oppo- 
site spin in a single orbit. Lithium, with three electrons, has two of 
opposite spins in its inner orbit, which is thus completely filled, and the 
third electron is in an orbit of greater radius. 

The energy-level picture for the electrons in the isolated lithium atom 
is of the general form shown in Fig. 3.2. There are two electrons with 
energy £1 and in the ground state there is one electron in state £2. The 
dotted levels above £ 2 represent permitted levels which the electron 
from £2 may occupy if excited. 

The energy-level system for a more complex multi-electron atom 
might have the general character shown in Fig. 3.3, where £1, £ 2 , and 
£3 are completely filled levels, and £ 4 is the outermost shell or level 
which may be only partly filled. 

The dotted levels are unoccupied levels into which electrons from £4 
may be excited. Apart from the innermost shell, E h the other levels 



QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



19 



contain more than two electrons (e.g. eight) when they are completely 
filled. Energy levels £2 and £3 should thus each more properly be 
shown as several very closely spaced full lines, indicating that there is 
really a set of filled sub-levels. 

Splitting of Energy Levels 

If two identical atoms are brought close together, then they react upon 
each other and the energy-level picture is modified. In the new system, 
now containing two atoms coupled together, there is a tendency for 
each of the original permitted electron energy levels to be split into two. 
The amount of splitting depends upon the closeness of the atoms and 
the phenomenon is analogous to the coupling of two identical resonant 
electrical circuits to give a system with a double-humped frequency 
response the precise shape of which depends on the tightness of the 
coupling. 

This type of coupling between atoms is responsible for the multi- 
plicity of energy levels in molecules and the much greater complexity of 
their spectra compared with those of simple atoms. Transitions be- 
tween closely spaced molecular energy levels give absorption and emis- 
sion effects at microwave frequencies which have been discussed 
elsewhere. 

In a solid crystal there are very many atoms close enough together to 
interact, and the single energy levels of the original atoms are broadened 
into wide bands of closely spaced levels. 



ENERGY 



DISTANCE 




Fig. 3.4 



Band System of Solids 

If we consider a one-dimensional crystal composed of a single row of 
atoms, then the potential energy of an electron has the periodic form 
plotted as a heavy line in Fig. 3.4 due to the positively charged nuclei. 
The horizontal lines shown in the figure are the electron energy levels 
in the isolated atom, full lines indicating occupied levels and dotted 
lines vacant levels. 



20 



LOW NOISE ELECTRONICS 



This much simplified view of the energy levels in the solid suggests 
that those with energies E\ and £2 are confined to the immediate 
vicinity of the nuclei in the potential troughs, while those possessing 
energy £3 can exist anywhere in the crystal. The electrons with energy 
£3 are free to wander at random through the crystal lattice — they are 
the "free" electrons responsible for electrical conduction in early 
classical solid state theory. 

A later, more comprehensive, theory takes into account the widening 
of the individual atomic levels into bands due to close packing. This 
theory explains most of the phenomena of current interest and we shall 
consider it below. 



ENERGY 






SEPARATION d 

Fig. 3.5 

Let us first consider two atoms separated by a distance d, each atom 
possessing the electron energy structure shown simplified as four levels, 
on the right-hand side of Fig. 3.5, for large values of d. We shall further 
assume that the element considered has levels £1, £ 2 , and £3 fully 
occupied. 

£1 thus contains two electrons of opposite spins, and £4 is an un- 
occupied level to which an electron from £3 could have gone if the 
atom had been excited. It has previously been mentioned that filled 
higher levels like £ 2 and £3 should be represented by a set of very close 
levels each containing two electrons of opposite spin. These sub-levels 
are, however, so close together that it is customary to show the energy 
of each shell as a single level. 

When the two atoms are brought close together each individual 
energy level tends to split into two, the higher levels being affected first 
as the atoms approach and the amount of splitting increasing as the 
separation decreases (Fig. 3.5). 






QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



21 



If the number of atoms reacting with each other is N and the average 
separation is d, then each of the single levels splits into TV levels. An 
energy-level picture similar to Fig. 3.6 (i) would be obtained with the total 
separation of the N levels into which each original atomic level splits, 
increasing as the average atomic spacing decreases. 

If the actual separation of the atoms in a particular crystalline solid 
is a (Fig. 3.6 (i)), then the electron energy level structure for that 
crystal will be as shown in Fig. 3.6 (ii). Each of the original levels of the 
isolated atom has been split into N levels, the higher bands now being 
of considerable width. 



ENERGY 




e: 2 

E1 



-~-Q~~ 



SEPARATION 

(i) 



(ii) 



Fio. 3.6 



In the isolated atom, £1, £2, and £3 were completely filled levels, so 
the corresponding bands in the solid state are also completely rilled. 
In particular IN electrons completely fill the N levels of the band £3 
which is separated by a considerable energy gap from the band of vacant 
levels above it. These electrons are thus unable to accept a small 
amount of extra energy from a battery connected to the material be- 
cause there are no permitted vacant levels near by. Since they cannot 
receive a small energy increment they cannot constitute a flow of 
current and the material is an insulator. 

If in the original atom the level £3 had been occupied by one instead 
of two electrons, then the N levels of the band £3 in the solid would 
contain only N electrons. These electrons would tend to fill the band 
from the bottom upwards, so that £3 would consist approximately of 
Nj2 filled levels with Nj2 vacant levels immediately above them. 

The electrons in the lower half of £3 would now be able to receive 



22 



LOW NOISE ELECTRONICS 



a small increment of energy from an applied electric field and the 
material would be a conductor of electricity. 

Overlapping Bands 

The implication of the previous section is that an element will only be a 
conductor in the solid state if the highest electron energy level of the 
atom is incomplete, e.g. it contains one and not two electrons, so that 
the band evolved from it has only half the positions occupied. There is, 
however, a way in which the uppermost band of a crystal may be only 
partly filled even though the top energy level of the isolated atom is 
completely occupied. 

In certain elements the levels split into wider bands than those in- 
dicated in Fig. 3.6 (i). At the atomic separation, which actually occurs 
in the crystal, all the vacant excited energy levels merge and this com- 
pletely vacant band overlaps the completely filled band evolved from 
the highest fully occupied energy level. 

The energy-level picture for the crystal will now have the form shown 
in Fig. 3.7 where the highest band, now partly filled, is due to an overlap 



E 3 iE 4 



Fig. 3.7 

between a filled and a vacant band. Note that where there are in- 
sufficient electrons to occupy the band completely, the levels are filled 
from the bottom up. 

Conductors and Insulators 

As long as we are only concerned with the electrical properties of the 
solid, then we need only consider the highest of the filled bands and the 
vacant band that lies above it. The energy-level pictures for all pure 
crystals then fall into one of the two classes shown in Fig. 3.8. 

When there is no gap between the filled and vacant bands the material 
is a conductor, and when there is a forbidden energy gap between them 



QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



23 



the material is an insulator because a small increment of energy is in- 
sufficient to take an electron from the filled valence band into a vacant 
level. 

It is worth noting that energy-level diagrams, unless otherwise stated, 
are drawn for zero degrees absolute. At any real temperature an 






CONDUCTOR 



INSULATOR 



Fio. 3.8 



electron may be thermally excited upwards into a slightly higher energy 
level if one is available. Thus some of the electrons near the top of the 
filled band in the conductor will be excited upwards into nearby vacant 
levels, leaving vacancies in the levels they previously occupied. Such 
thermal excitation will not occur in the insulator because of the width 
of the forbidden energy gap. 





n-TVPE 



p-TYPE 



FILLED IMPURITY LEVEL 
VACANT IMPURITY LEVEL 

Fig. 3.9 



Impurity Semiconductors 

If a material in the pure form has the characteristic insulator struc- 
ture, then the addition of small amounts of suitable impurity may 
change it into a semiconductor. Dependent on the kind of impurity 
added, the "doped" material may become either an «-type or a p-typt 
semiconductor (Fig. 3.9). 
In the n-type semiconductor the impurity gives rise to an extra 

B 



24 




LOW NOISE ELECTRONICS 



occupied electron energy level up near the conduction band. Such an 
effect is achieved by doping a tetra-valent element like silicon with a 
group five impurity like arsenic, or by preparing a group 3-group 5 
compound like gallium phosphide so that it is rich in the group 5 
element. 

A classical view of the conduction process is that four of the valence 
electrons of the impurity atom are used to bind it into the tetra-valent 
crystal lattice, leaving the fifth as a "free" electron which allows con- 
duction to take place. The energy level explanation is a little different. 

The extra energy level due to the impurity atom will not be a general 
property of the crystal but will represent a situation existing only in the 
vicinity of an impurity atom. For this reason the impurity level is 
drawn as a short line to indicate that it is localised. 

The situation in Fig. 3.9 is, however, that which occurs at zero degrees 
absolute, and at any higher temperature some of the impurity electrons 
will be excited up into the nearby conduction band. The filled or donor 
impurity level is usually so close to the conduction band that at room 
temperature all the impurity electrons will be excited into it. Once 
excited thermally into the conduction band the impurity electrons can 
exist anywhere in the crystal. They can also accept a small increment of 
energy, so the material will conduct electricity. The conductivity will 
depend on the number of electrons available in the conduction band, 
and since at room temperature all the impurity electrons are excited into 
this band, the conductivity will be proportional to on the number of 
impurity atoms present, i.e. the percentage doping. 

A p-typQ semiconductor (Fig. 3.9) may be obtained if a tetra-valent 
material is doped with a trivalent impurity, e.g. indium, or if a group 
3-group 5 compound is prepared in a form rich in the group 3 element. 
A vacant, or acceptor, impurity level or "hole" is produced down near 
the filled band and the energy level situation at zero degrees absolute is 
as shown. 

At higher temperatures the hole is excited downwards into the valence 
band and is now free to exist anywhere in the crystal. It behaves as 
though it were charged positively and is responsible for the conduction 
in the /J-type impurity semiconductor. 

At 0° K each hole is localised in the vicinity of its impurity atom, but 
at higher temperatures one of the electrons from a neighbouring atom 
of the host material will occupy the hole, leaving a new hole by the host 
atom. This new hole will in turn be occupied by an electron from 
another atom, so that there appears to be a hole wandering at random 
through the material. The original impurity atom is now negatively 
charged but fixed in the crystal, while the hole wandering through the 



QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



crystal is effectively positively charged because its motion in one 
direction really represents the movement of an electron the same 
distance in the opposite direction. 

The electrical situation at room temperature in a />-type semicon- 
ductor consists of a collection of stationary negatively charged impurity 
atoms distributed throughout the crystal and an equal number of 
positively charged holes wandering at random, so that the nett space 
charge is zero. 

In the n-type semiconductor the impurity atoms become positively 
charged and are fixed in position, with electrons wandering at random, 
the nett space charge again being zero. 

The behaviour of the p- and H-type semiconductor in contact is most 
important technologically. Devices such as rectifiers, transistors, tunnel 
diodes, and varactor diodes for parametric amplifiers all depend on the 
properties of the p-n junction. 

Intrinsic Semiconductors 

Certain solids possess an energy-level picture (Fig. 3.10 (i)) which has 
the characteristic insulator form, with the empty conduction band 
separated from the filled valence band by a forbidden energy gap. In 
this case, however, the energy gap is small and at temperatures above 



I 

SMALL 
6AP 

I 






00 






© ® 



(i) QD 

Fro. 3.10 

0° K it is possible for an electron to be excited thermally across the gap 
into the conduction band, leaving a hole in the valence band. These 
intrinsic holes and electrons (Fig. 3.10 (ii)) can now act as charge 
carriers and the material will conduct electricity. 

At any temperature hole-electron pairs will be produced at a certain 
rate in an intrinsic semiconductor and will also disappear at a certain 
rate when holes and electrons come together in the course of their 
random motion. At a given temperature there will be a certain number 
of charge carriers available, and hence a certain conductivity which 
increases with temperature. 



26 LOW NOISE ELECTRONICS 

Germanium, which is commonly used as a host material to which 
impurities are added to make impurity semiconductors, is also an 
intrinsic semiconductor. In such a germanium-based impurity semi- 
conductor a significant amount of intrinsic current flows at room 
temperature. Devices like rectifiers and transistors depend upon im- 
purity effects, and the presence of intrinsic current is a disadvantage, so 
that germanium transistors cannot be operated at temperatures above 
70° C. 

The unwanted effects of intrinsic semieonduction can be reduced by 
using a substance like silicon, with a larger energy gap, as the basic 
material. 



The p-n Junction 

It is possible to prepare a single crystal of a semiconductor so that one 
end of it is doped with p-type impurity atoms and the other end with 
n-type impurity atoms. The region in which the change from p- to n- 
type material occurs is called the junction and its character is different 
from that of the rest of the material. We shall consider the electrical 
character of a junction where the change from p to n takes place sharply 
but the same general properties arise in the graded junction where the 
transition is not so sharp. 



p-TYPE 



A F B 



n-TYPE 

ENERGY 




P-TYPE 
A 



n-TYPE 

F B 



r© 4 © © © 

i*© + © © © 

V© © © 

i*0 + © © © 



© © ©"©" 
© © ©~© 
© ® ©"© 
© ©"© ©~ 



ENERGY C G D 

(i) 



©1 IONISED IMPURITY 
©J ATOMS (FIXED) 

~\ ELECTRONS & HOLES 
+ J (MOVING) 

(ii) 



Fig. 3.11 



Figure 3.11 (i) shows the energy-level picture at a sharp discontinuity 
FG between a p- and an n-type semiconductor. The horizontal dotted 
lines show the bottom of the vacant band and the top of the filled band 
in the positions they would have occupied if there had been no contact 
between the n- and the /^-regions of the crystal. 



QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



27 






At room temperature the impurity electrons in the n-type material 
and the holes in thejc-type material will have been excited into the vacant 
band and the filled band respectively. Within these bands the electrons 
and holes will be distributed exponentially with energy as indicated by 
the shaded distribution curves at X and Y. 

In Fig. 3.11 (ii) is shown the distribution of charge in the material: 
to the left of A and to the right of B is seen the charge distribution which 
would have existed in the/?- and the n-type materials respectively if there 
had been no contact. In each case the charge due to the fixed ionised 
impurity atoms is just neutralised by the opposite charge of the holes 
or electrons which move at random through the material. 

If we now imagine the contact to be made, then the electrons from the 
n-type material will diffuse into the />-type, and holes from the /7-type 
will diffuse into the n-type. If this were only a diffusion process, then 
it would continue until the concentration of holes and electrons were 
constant throughout the crystal. 

In fact, however, the electrons which leave the n-type material leave 
it positively charged, and the holes which leave the p-type material 
leave it negatively charged. They diffuse into the material in which 
they constitute minority charge carriers (e.g. electrons in /J-type) until 
they meet a carrier of the opposite sign. When an electron and a hole 
meet they combine and both charge carriers are removed from the 
electrical conduction process. The immediate vicinity of the junction, 
say between A and B, contains no holes or electrons and is an insulator 
across which there exists a potential gradient due to the space charge of 
the ionised impurity atoms, now no longer neutralised by free carriers. 
The region between A and B is called the depletion layer. 

The energy levels on each side of the junction move vertically with 
respect to each other because of the potential difference now existing 
between the/>- and the n-type material. The final situation is shown by 
the full lines in Fig. 3.11 (i). Electrons have passed from n to p and 
holes in the opposite direction until a sufficiently great potential differ- 
ence is established to prevent any further diffusion. When this has 
happened the hill in the energy level diagram is just high enough to 
prevent the most energetic electron in the n-type material — or hole in 
the/Kype — from crossing the junction. The peak of the distribution 
curve V (or W) just reaches the top of the potential hill. 

If an external voltage is applied to the junction, then the height of the 
hill is either raised or lowered and rectification occurs. If the external 
voltage is applied in the forward direction with the positive terminal to 
the p-type material, then the hill will be lowered and the tops of the 
distribution curves will project over it. Electrons will be injected from 



28 



LOW NOISE ELECTRONICS 



n into p and holes from p into n, making a large forward current. With 
the external voltage applied the other way round, the hill is increased 
in size and no current flows, although in a real rectifier there is some 
reverse current due to intrinsic charge carriers. 

Tunnel Diode 

If enormously high concentrations of impurities are used in a p-n 
junction the potential hill formed without external bias voltage becomes 
so high that the bottom of the conduction band of the n-type material, 
with its impurity electrons, is depressed below the top of the valence 
band of the />-type, with its holes. The electrons are separated from 
permitted vacant states at the same level in the other material only by 
the depletion layer which is so thin that quantum mechanical tunnelling 
occurs through it. 

As the forward bias is increased the conduction and valence bands of 
the two materials no longer overlap and the tunnel current is replaced 
by ordinary forward current. Such a diode is called a tunnel or Esaki 
diode and the I-V characteristic has a valley in it with a negative slope 
which may be used for amplification or oscillation. Such diodes are 
likely to be developed as low noise amplifiers at frequencies up to many 
Gc/s. 

Gunn Effect and Other Properties 

The greatest technological application of semiconduction has been the 
exploitation of p-n junction properties in the transistor, where a for- 
ward and a reverse biased junction are made to interact. The tunnel 
diode mentioned above represents a simplification, where a single 
junction may be made to give a negative resistance effect and thus used 
as an amplifier or oscillator. Similar oscillatory effects have been 
observed in other types of diode operated in special ways. 

A great simplification in electronic devices might be achieved if such 
effects could be obtained in semiconductors without the need for special 
metallic contacts, or junctions between p- and n-type regions. In 
particular, at frequencies in the few Gc/s region, which is so important 
in modern communications, the mechanical tolerances on manu- 
factured devices like klystrons and transistors are becoming very 
stringent, but at the same time the dimensions involved are becoming 
closer to those naturally available in crystal structures. If bulk ampli- 
fiers and oscillators can be found, then they are likely to replace 
elaborate manufactured devices because they will be easier and cheaper 
to make, more efficient at heat dissipation, faster in response time, and 
more robust. 



QUANTUM PICTURE OF ELECTRONS IN SOLIDS 



29 






The Gunn effect 1 is an important example of a bulk phenomenon 
which has already led to a practical device. 

If an electric field is applied to a piece of gallium arsenide then a 
current flows. If this field is now increased the current also increases, as 
would be expected. But if the field is increased still further, then the 
mean current flowing decreases, thus showing a negative resistance 
effect which in principle might be used to give amplification or oscilla- 
tion. There is, however, an extra feature to the current in the negative 
resistance region. In general, it shows very rapid fluctuations in value 
and in particular, if the length of the gallium arsenide across which the 
field is applied is made small, then the fluctuation in current occurs at a 
particular frequency in the Gc/s region. 

The effects observed depend principally on the fact that the gallium 
arsenide has an energy level structure like that of Fig. 4.3 (p. 37). 
Electrons will normally be found in the lower valley of the conduction 
band, but they may transfer to the higher valley if given sufficient 
energy by, for example, a high electric field. It so happens that the 
effective mass of these "hot electrons" in the higher energy valley is 
greater than that of those in the lower valley, and hence the mobility of 
the hot electrons is less. If a high field causes electrons to move into 
the higher energy valley the current in the material will actually fall, 
because of the reduced mobility, and a negative resistance effect will 
occur. 

When the field is high enough to produce hot electrons its distribution 
throughout the material no longer remains uniform. The fall in 
mobility due to the production of hot electrons at one point gives in- 
creased resistivity and hence a rise in the field in that region which will 
produce even more hot electrons and a further rise in the field. Since 
the total voltage available across the specimen is fixed, the field in the 
normal region of the gallium arsenide will fall, and so will the current. 

The high field region is formed first at the cathode. It builds up 
rapidly and then moves across the specimen to the anode where it dis- 
appears. The field in the specimen then reverts to the uniform state for a 
short time until another instability builds up at the cathode and sets out 
to the anode. 

If the specimen is long, disturbances set out from the cathode one 
after the other and the current fluctuations have a largely random 
character. If, however, the specimen is made short, only one dis- 
turbance at a time crosses from cathode to anode and the current 
fluctuations have a regular pattern at a frequency inversely proportional 
to the specimen thickness which is typically about 0*1 mm. 

1 "The Gunn Effect", Gunn, International Science and Technology, October 1965. 



LOW NOISE ELECTRONICS 



If one of the terminals is connected to the inner of a coaxial cavity 
then R.F. oscillations are produced in the cavity and both pulsed and 
c.w. operation have been achieved. 2 An experimental oscillator giving 
several milliwatts of c.w. power at 1 Gc/s has been reported 3 with 
prospects of improved performance in the near future. 

2 "Continuous Microwave Oscillations of Current in GaAs", Gunn et at., I.B.M. 
Journal, November 1964. 

3 Electronic Engineering, September 1965. 






4 

The Interaction of Photons, Phonons, 
and Electrons 



in the previous chapter we have examined what determines whether 
a substance is a conductor of electricity. The discussion was based on 
the energy-level structure of the solid and it was only necessary to con- 
sider the most energetic of the electrons present. The criterion for 
conductivity was found to be that these electrons should be able to 
accept small increments of energy from an applied electric field. 

We have hardly mentioned what happens to the electrons once they 
are moving through the material under the influence of an electric field, 
although experience of the heating effect of an electric current suggests 
that they give energy to the bulk of the solid material. 

It is thus possible for energy to be transferred from an electron to the 
crystal lattice. The converse process is also possible, i.e. energy may be 
transferred from the lattice to an electron. This converse process has 
already been met as the thermal excitation of electrons, e.g. across the 
energy gap in the intrinsic semiconductor (p. 25), or from a donor 
impurity level into the conduction band in the n-type impurity semi- 
conductor. 

Joule heating and thermal excitation both involve the interchange of 
energy between an electron in the material and the crystal lattice — both 
are "internal" energy processes. 

There are also important "external" energy processes (e.g. in photo- 
conductors, phosphors, and lasers) where an electron is excited by 
radiation entering the substance from outside, or, conversely, where an 
electron falls from a high to a lower energy level and emits radiation 
which escapes from the material. 

The external energy interchanges are well known to be quantum 
processes involving bundles of energy E — hf, called quanta or photons. 
These interchanges are often known as photon-electron interactions. 

Similarly, the internal energy interchanges are quantum processes. 



32 



LOW NOISE ELECTRONICS 



The quantum of lattice energy involved is known as a phonon, by 
analogy with the photon, and the processes as phonon-electron inter- 
actions. 

In this chapter we consider some of the properties of phonons, 
photons, and electrons, and the ways in which they interact. 

Electrical Resistance 

If an electric field, £, is applied to a free electron, then it experiences a 
force Ee and a constant acceleration Eejm. The electron velocity thus 
increases continuously, as docs the electron's energy, as long as it re- 
mains in the field. 

This is certainly not the case when a current flows in a conductor due 
to an applied field. In this case the electrons appear to move with a 
constant drift velocity, and their mobility \i is defined as the average 
velocity acquired when a unit electric field is applied. There must 
therefore be some additional mechanism involved which extracts energy 
from the electrons as they move through the conductor. This extra 
mechanism is, in fact, the collision of the conduction electrons with the 
crystal lattice. The average time that the electron moves freely between 
collisions is sometimes called the relaxation time t. 

Mobility and relaxation time are related by the expression 

erz 

\k = — 
m 

where m is more properly the effective mass of the electron which may 
differ from the true mass because of the special conditions under which 
the electron exists in the solid. 

The conductivity of the material is given by neii, where « is the number 
of conduction electrons per unit volume. 

In a material where the conductivity is wholly or partially due to 
holes the same expressions may be used, but the values of y. and m will, 
in general, differ from those used for electrons. 

Collisions with the Lattice 

The simple view of the electron as a solid particle "colliding" with the 
equally solid ions of the crystal lattice gives a qualitative idea of the 
origin and nature of electrical resistance. It is, however, unable to 
account for all conductivity phenomena, especially the variation with 
temperature. In particular superconduction, the complete disappear- 
ance of resistance in certain materials at very low temperatures, seems 
to imply the disappearance of the crystal lattice, which is clearly absurd. 
To get a full explanation of the conduction process the electron must 



THE INTERACTION OF PHOTONS, PHONONS, AND ELECTRONS 



33 






be regarded as a wave packet interacting with the periodic potential field 
of the lattice. Such a wave-mechanical treatment can be found in many 
advanced textbooks on the solid state. 1 

In the wave-mechanical treatment a "collision" becomes the random 
scattering of the electron wave and it is shown that a perfect periodic 
potential structure does not affect the propagation. It is only when the 
periodicity of the potential becomes disturbed that scattering occurs 
and electrical resistance appears. 

Such disturbances of the perfect periodicity of the lattice occur as a 
result of thermal vibration at any temperature above the absolute zero. 
These vibrations are, in fact, elastic waves propagated in the solid at 
the velocity of sound, which is much less than the velocity of the 
electron waves. Consequently, the electron wave effectively passes 
through a stationary lattice with its regularity instantaneously distorted 
by the elastic wave. 

Disturbances of the periodicity of the lattice, and hence scattering, 
may be due not only to thermal vibration but to distortion caused by 
foreign atoms occurring either accidentally in impure materials, or by 
design in alloys. 

Material of the highest purity must be prepared if the best conduc- 
tivity is to be obtained, e.g. 0-1% of phosphorus impurity in copper 
reduces the conductivity by about 50%. Physical distortion in the 
lattice can be caused by strain due to working the material, and the 
conductivity may be improved several per cent by annealing. If the 
material is prepared as an alloy for reasons of mechanical strength, then 
a fall in conductivity must usually be accepted, e.g. a copper-cadmium 
alloy containing a little less than 1% cadmium has a conductivity just 
less than 90% of that of pure copper. 

Phonons 

There are a number of phenomena in physics which can be described 
either in terms of waves or particles, although in any particular problem 
one approach may be more suitable than the other. 

The controversy about the wave and corpuscular nature of light was a 
lengthy historical argument and the explanations of interference effects 
in terms of wave properties, and of the photo-electric effect in terms of 
photons, are both well known. The photo-electric effect is concerned 
with the amount of energy E = hf in one photon, but by using the 
Einstein E — mc 2 relation it is possible to ascribe to the photon an 
effective mass and momentum which must be used to explain the 
Compton scattering of X-rays (p. 42). 

1 Wave Mechanics of Crystalline Solids, R, A. Smith, Chapman & Hall. 



34 



LOW NOISE ELECTRONICS 



In the case of the electron its particle-like properties of mass and 
momentum were long known and used before its dual character was 
also established. Its wave properties are the basis of the wave mechanics 
which is a powerful tool in the investigation of the solid state, and one 
of the results of which we have already used in noting above that a 
perfectly periodic crystal lattice offers no resistance to electric current. 

One of the principal sources of electrical resistance is the distortion 
of the lattice due to thermal vibrations, and this provides us with a third 
example of duality. 

The distortions travel through the material as elastic waves and this 
vibrational energy is quantised, just as light energy travelling as an 
electromagnetic wave is quantised. By analogy with the photon we can 
consider the passage of vibrational energy through the solid as due to 
phonons moving through the material at the velocity of sound. The 
phonon is thus a quantum of lattice energy and, like the photon, it will 
possess momentum and mass. 

Electrical resistance can now be reconsidered as due to the collision 
of electrons with phonons (and with impurities). As the temperature 
rises, increased atomic vibrations can be represented by increasing 
numbers of phonons flowing through the crystal. More collisions with 
phonons will now occur, giving reduced relaxation time and mobility 
and hence an increase in resistance with temperature. 

We shall see later that phonon momentum may be important in 
certain electron-photon interactions in solids. 

Thermal Conductivity 

If one part of a substance is maintained at a high temperature and 
another at a low temperature, then energy flows from the hot to the 
cold region. It may be convenient to think of the energy being trans- 
ported by particles, and these "particles" will be phonons in non-metallic 
solids, and phonons and electrons in metals. 

The thermal conductivity is proportional to the mean free path of the 
particles, and the mean free path of electrons is much greater than that 
of phonons. Consequently, the thermal conductivity of non-metals is 
very low compared with that of metals, although the presence of foreign 
atoms, either as impurity or in alloys, sharply reduces metallic thermal 
conductivity. 

There is obviously great similarity between the mechanisms of 
thermal and electrical conductivity, and in the case of metals this is 
summed up in the Wiedemann-Franz law which suggests, with fair 
accuracy, that the ratio of the thermal to the electrical conductivity is the 
same for all metals at a given temperature. 



THE INTERACTION OF PHOTONS, PHONONS, AND ELECTRONS 35 

Electron-Photon Interactions 

The phenomena of electrical and thermal conductivity discussed above 
are largely due to electrons and phonons, and the collisions between 
them. In solid state electronics there is often emission of quanta of 
radiation when electrons undergo energy changes. Such electron- 
photon interactions in solid semiconductors concern us particularly 
because of their occurrence in the crystal injection laser, and are of 
general interest in the emission of light by phosphors. The converse 
processes involving absorption of quanta of radiation are perhaps of 
less interest in practical electronics, although the study of these pro- 
cesses yields valuable information about the solid state. 

The simplest possible approach to the emission of a photon is that 
which deals only in conservation of energy. 

In Fig. 4. \ is shown the energy-level structure for a semiconductor 



© 



© 



E=hf 






Fig. 4.1 

with a hole and an electron, produced by some sort of earlier excitation, 
in the valence and conduction bands respectively. The hole and the 
electron will be wandering at random through the material. If they 
meet then they will recombine, disappear as charge carriers, and a 
photon of radiation will be emitted. 

Assuming the electron to fall from the bottom of the conduction band 
into a hole at the top of the valence band, then the frequency of the 
radiation emitted is given by E = hf, where E is the energy gap in the 
material This energy gap must be large if visible light is to be emitted. 

Such a simple process is called direct recombination, but in many 
cases the recombination is complicated by extra levels due to imper- 
fections or impurities. 

In Fig. 4.2 (i) an impurity level, in this case a vacant level, is shown 
near the edge of the conduction band. 

The recombination is now a two-stage process illustrated in Fig. 
4.2 (ii) and (iii). 

If the electron arrives at one of the sites in the crystal where there is 



36 



LOW NOISE ELECTRONICS 



an impurity level, then it falls into that level and remains fixed in posi- 
tion in the material (ii). Subsequently, one of the holes moving through 
the material will arrive at this site and be trapped by the temporarily 
filled impurity level (iii). The level is now in its original state and again 
available to assist in the recombination of a free hole and electron. 

If both processes, i.e. the original capture of an electron and the sub- 
sequent capture of a hole, are carried out with equal probability by the 
impurity level then it is called a recombination centre. A similar re- 
combination centre may be due to the existence of a filled instead of a 
vacant impurity level, but in this case the first process will be the 
capture of a hole and the second the capture of an electron. 






If any impurity centre does not perform each of the two processes of 
recombination with equal probability, then it may be known as a trap. 
Thus a vacant impurity level may readily capture an electron, but, when 
filled, have a much lower probability of completing the recombination 
by capturing a hole. Such an impurity level would be known as an 
electron trap. 

Usually the free holes and electrons produced in a material by some 
form of excitation will recombine fairly quickly, emitting the excess 
energy at the appropriate frequency. But if trapping centres exist in the 
material, say electron traps, then some of the excited electrons will be 
trapped and held for a time in a state of comparatively high energy. 
They will eventually recombine with free holes and emit energy, but this 
may not occur until some time after the original excitation is removed. 
The material will thus behave as a phosphor with "after-glow" pro- 
perties. 

Features of the material which determine the way in which energy is 
emitted will be the total width of the energy gap, and whether the 
impurity level is near the conduction or valence band (shallow), or 
approximately in the middle of the gap (deep). 

A complete description of the possible recombination processes and 
an explanation of the character of the energy emitted would obviously 



THE INTERACTION OF PHOTONS, PHONONS, AND ELECTRONS 37 

be very complex. Indeed, the details of the processes involved are not 
yet fully understood. One important consideration must, however, be 
added to the simple conservation of energy principle used above. 

The principle that momentum as well as energy must be preserved in 
crystal processes is fundamental theoretically, and has important 
implications in applied quantum electronics. 

In discussing the conservation of crystal momentum in electron- 
photon processes in semiconductors it is useful to consider the energy- 
momentum diagram for a crystal. A very simple theoretical example in 
one dimension is shown in Fig. 4.3. 



ENERGY 




TRANSITION 



ENERGY 
GAP 




VALENCE 
BAND 



CRYSTAL MOMENTUM 

Fig. 4.3 

Here the upper line represents the conduction band and the lower line 
the valence band, and there is a forbidden energy gap between them. 
Electrons are most likely to be found at the conduction band minimum 
and holes at the valence band maximum. Since the momentum of a 
photon in or near the visible spectrum is so small there must be negligible 
change in crystal momentum when an electron falls from the conduction 
band to the valence band and emits a photon of energy. 

in the diagram shown, such a transition is readily possible and the 
material will be known as a direct gap semiconductor (e.g. gallium 
arsenide). 

More often the energy-momentum diagram will have the form 
shown in Fig. 4.4, where in general the transition of an electron across 
the gap will involve a considerable change in crystal -momentum. The 
electron can only make the direct transition shown, and radiate a 
photon of energy, if the extra momentum is removed simultaneously as a 
phonon. This process is not very probable and in such materials the 
light-emitting transition of an electron across the gap will be indirect 
and will make use of an impurity level in the gap. The impurity centre 
must be such that it absorbs a phonon of the right momentum, but it 



38 



LOW NOISE ELECTRONICS 



must not be so deep in the energy gap that the subsequent transition 
does not produce visible light, i.e. if the material is required as a 
crystal lamp or injection laser (p. 129), then luminescent impurity centres 
of this type must be encouraged, and other, non-radiative centres 
avoided. 

Indirect gap materials are silicon and germanium, but their energy 
gaps are so small that light is not emitted. A luminescent indirect gap 
materia], such as gallium phosphide, will emit light of a colour which 
depends upon the type of added impurity. 



ENERGY 



CONDUCTION 
BAND 




CRYSTAL MOMENTUM 



Fig. 4.4 



A radiative transfer of an electron from the conduction to the 
valence band of a direct gap semiconductor is an electron-photon 
interaction, whereas the same process in an indirect gap material usually 
proceeds by way of an impurity centre and can be considered as an 
electron-phonon-photon interaction. 

Tn electrical conduction the phonon is a crude spoiler, limiting by 
collisions the velocity acquired by the electrons, but in luminescence it 
plays a useful role by allowing radiative transfers to occur in indirect 
gap materials. We shall see elsewhere (p. 59) another example of the 
phonon being useful: in the phonon maser. 

Absorption — Excitons 

The previous section dealing with electron-photon interactions assumed 
that excitation of electrons into the conduction band had already taken 
place. It then considered how radiative transitions of these electrons 
back across the gap to the valence band would occur. The converse 
process of absorption of a photon of radiation to produce an electron 
in the conduction band and a hole in the valence band must obey the 
same rules of conservation of energy and momentum. As with lumine- 
scence, the way in which the electron transition takes place will be 



THE INTERACTION OF PHOTONS, PHONONS, AND ELECTRONS 39 



greatly influenced by the nature of the energy-momentum diagram of 
the material and may require the emission or absorption of a phonon. 

Apart from the fact that radiation absorption in solids follows the 
same rules as emission it is of much less practical interest to the elec- 
tronics engineer. Much of the fundamental knowledge of the solid state, 
however, comes from absorption experiments carried out in the physics 
laboratory. 

The width of the energy gap in a semiconductor can be determined by 
observing the maximum light wavelength which will cause conductivity. 
Such experiments are best carried out at low temperatures where there 
will be few phonons to collide with electrons in the valence band and 
excite them into the conduction band where they will produce un- 
wanted intrinsic conduction effects which might mask the photocon- 
ductivity which is to be observed. 

When such experiments were carried out with light of gradually in- 
creased wavelength it was found that for a given material there was a 
certain wavelength at which the absorption suddenly fell. This maxi- 
mum wavelength showed the width of the energy gap. But at certain 
still longer wavelengths there was a significant amount of absorption 
indicating the presence of extra levels in the gap. Such levels could be 
due to impurities, and indeed this absorption technique can be used to 
determine the position of the impurity levels. If impurity levels are the 
cause of the discrete absorption at longer wavelengths, then it will be 
accompanied by photoconductivity due to the mobile charges produced 
in either the conduction or the valence band. 

There is, however, another phenomenon which produces selective 
absorption at certain wavelengths longer than the expected maximum 
value. This is known as the exciton process. 

In the simple case of the excitation of an electron from the valence 
to the conduction band a hole is produced in the valence band. 
Ordinarily the hole and the electron move off in different directions 
through the material as quite separate carriers of charge and thus con- 
tribute to the electrical conductivity. In an analogous way, a hydrogen 
atom given enough energy would be ionised and the electron and the 
positive nucleus would go their different ways. 

If, however, the hydrogen atom had been given less than the energy 
required to produce ionisation, then it might have been merely raised to 
one of the discrete set of excited energy levels. In a similar fashion a 
solid may absorb a photon which is of insufficient energy to excite an 
electron across the gap and thus produce a separate hole and electron. 
The photon may, however, create a hole-electron pair still bound to- 
gether, but in an excited state like the hydrogen atom. This bound 



40 LOW NOISE ELECTRONICS 

hole-electron pair may move through the material and is called an 
exciton. 

The exciton gives rise to extra energy levels which lie within the energy 
gap of the material. They are not, however, like vacant impurity levels 
because, although the electron may be excited into one of them, it is 
still bound to the hole and no conductivity results. 

Excitons may thus contribute to the absorption or omission of energy 
in a solid but not to the conductivity. 

More Complex Interactions 

There is still much to be learned about the solid state and the inter- 
actions between electrons, photons, phonons, 2 and excitons discussed 
briefly above are some of the simplest which are needed to explain 
important general principles. The detailed explanation of many 
phenomena involves more complex interactions, perhaps the emission 
of radiation by an exciton, or the emission of many phonons by an 
excited trap. 

Perhaps the most important of these more complex interactions are 
the so-called non-linear photon processes which are now being dis- 
covered and investigated using the very high energy fluxes available from 
lasers. Typical of these processes is the excitation of an electron from 
the valence to the conduction band by the simultaneous absorption of 
two or more photons, not necessarily of the same frequency. Another 
non-linear process is the simultaneous absorption of two or more 
identical photons and the emission of a single photon at a harmonic of 
the original photon frequency. Thus the techniques of frequency 
doubling or harmonic generation common in conventional tuned cir- 
cuit electronics may be applied at the highest radio frequencies and in 
the optical region. 

Accounts of such new phenomena and applications of recently dis- 
covered ones appear regularly in the scientific journals. The periodicals 
quoted in the bibliography (p. 141) are typical of those which carry 
such reports. 

2 "Phonons and Phonon Interactions", B.A.K., published Benjamin (1964). 






Radiation and Matter — Einstein's 
Radiation Theory 



in the earlier discussion of the interactions between photons and 
electrons in solids it was found that the principle of conservation of 
energy was not enough and that conservation of momentum had also 
to be taken into account (p. 37). Nevertheless the momentum of the 
photon in these interactions was usually very small compared with that 
of the electron, and it was the momentum of a phonon which was 
relied upon to balance the momentum equation if required. 

In this chapter, however, we are concerned principally with the rela- 
tions between radiation and matter, and we shall find that the momen- 
tum associated with a photon of electro-magnetic radiation must be 
taken into account if certain important processes are to be explained. 
Some such processes of fundamental and historical importance are 
briefly dealt with below, but one of especial interest is considered more 
fully. This is Einstein's investigation of fundamental radiation pro- 
perties which led to the recognition of the phenomenon of stimulated 
emission and, eventually, to the use of this process in masers and lasers. 

Compton Effect 

The early years of the twentieth century saw the reopening of the old 
discussion about the dual nature of electromagnetic radiation. This 
time, however, the wave and corpuscular theories were not in conflict, 
but it was recognised that the choice of either to explain a given 
phenomenon was largely a matter of convenience. By 1924 the duality 
concept had been extended to matter by de Broglie, who suggested the 

association of a wavelength X = — with a moving particle. Conversely, 

at about the same time (1923), Compton explained a particular form 
of X-ray scattering by associating momentum with electromagnetic 
radiation. 



42 



LOW NOISE ELECTRONICS 



In certain cases the X-rays scattered from a material have a longer 
wavelength than the incident radiation and the change in wavelength dx 
depends on the scattering angle <f>. 

d X = — (1 — cos<£) 
mc 

Compton scattering (Fig. 5.1) is due to a photon of X-radiation 
striking an electron. The electron recoils with kinetic energy \mv z at an 
angle 9 to the path of the incident photon, and the scattered photon 
moves away at an angle <f> to the incident path with a diminished energy. 
The scattered radiation is of longer wavelength than the incident radia- 
tion, since hf = hf + $mv 2 , i.e. energy is conserved. 



SCATTERED PHOTON 



INCIDENT PHOTON 
E=hf 



RECOIL ELECTRON 

1 /2mV 2 
Fig. 5.1 

It is, however, possible to observe experimentally both the direction 
of the scattered X-radiation and that of the recoil electron. To explain 
the relation between and <f>, and to derive the relation between d\ and 
<j>, it is necessary to consider conservation of momentum as well as 
conservation of energy. 

Because of the equivalence E = mc 2 between mass and energy, a 
photon, moving with the velocity of light c, has an effective momentum 
hfjc. Two equations for the conservation of momentum may thus be 
derived 



— — — cos 4> + w cos 



¥' - . 

— sin * = mv sin 
c 



Combined with the energy equation hf = hf + \mv 2 these equations 
enable d\ = X' — X to be determined in terms of 0, i.e. 



*/X = — (1 — cos <f>) 

mc y T/ 



RADIATION AND MATTER— EINSTEIN'S RADIATION THEORY 



43 



Since dx is independent of wavelength, the fractional change in wave- 
length, which will determine how easily the effect may be observed, 
becomes greater as the wavelength decreases. In general, Compton 
scattering is only observed with the high energy, high momentum 
photons in the X-ray and y-ray region. 

Mossbauer Effect 

The Compton effect was an early demonstration of the momentum of 
radiation and was consequently very important theoretically, although 
of negligible practical importance. The Mossbauer effect is a very much 
more recent (1959) demonstration of the momentum of radiation and is 
thus intrinsically not of such great theoretical importance as the 
Compton effect. It is, however, of great practical importance because 
it provides a tool for experimental work of considerable interest in 
fundamental scientific theory. 

The nucleus of a radioactive element which emits y-rays only will 
have two energy states. A transition from the high state to the low state 
will involve the emission of a photon of y-radiation. If £2 and £1 are 
the energy levels, then the frequency of the y-radiation is given by 
E 2 — £1 = hf. The width of the spectral line emitted (cf. p. 15) will 
increase with the probability of such emission occurring, and a narrow 
y-ray line will hence be emitted by a radioactive material of relatively 
long half-life. 

In the case of y-radiation it is more common to define the photon 
energy than its frequency, and for a particular element Fee? the energy 
of the emitted y-radiation is about 14,000 eV and the line width about 
10-s eV. 

Suppose that the nucleus emitting the radiation is free to recoil, then 
in order that momentum be conserved the nucleus will move back- 
wards with momentum equal to that of the photon. There will be some 
energy associated with the nucleus as it recoils, and this much less 
energy will be available for radiation. The energy emitted will thus be 
hf instead of hf and the frequency of the radiation/' </. 

In the same way a nucleus could normally absorb energy hf and 
become excited from state £1 to state £2. But, if the nucleus is free to 
recoil on absorbing the photon of radiation, then the incident photon 
will have to possess rather more energy than hf in order to provide the 
recoil energy and the excitation energy £2 — £1. The frequency of the 
radiation absorbed by the nucleus will thus be/" where/" >/. 

In a material where the nuclei are free to recoil then the y-ray emission 
frequency differs from the absorption frequency by/" — /'. 

The energy associated with the recoil of the iron nucleus is about 



44 



LOW NOISE ELECTRONICS 



10~ 3 eV and the line width of the radiation only 10~ 8 eV, so that the 
absorption and emission spectra do not overlap. 

If, however, the nuclei are set in a solid crystal lattice, usually cooled, 
then there is negligible energy lost in recoil and the absorption and 
emission frequencies coincide. In such a case the absorption by un- 
excited nuclei of radiation emitted by excited ones can readily be 
detected. Because of the extremely narrow line width anything which 
causes the emission frequency to change slightly will result in a fall in 
absorption. 

Such a change in frequency can be caused by the Doppler effect, and 
relative velocities between source and absorber of just a few feet per 
hour can be detected. 

A change in frequency is predicted by the general theory of relativity 
if a photon moves in a gravitational field. Only the Mossbauer experi- 
ment is sensitive enough to detect this effect. If a source is placed at the 
top of a tower and the absorber at the bottom, then the change in 
absorption observed is of the correct order. More accurate experi- 
ments along these lines in the near future are expected to provide 
stringent tests for the relativity theory. 

Einstein Theory of Mechanism of Radiation 

The Maxwell curve for the distribution of velocities among gas mole- 
cules and the Planck curve for the spectral distribution of thermal 
radiation are very similar in shape. Wien had made use of this simi- 
larity in deriving his pre-quantum expression 



Ef=Pi( E f ) 



(A) 



where £/ is the energy density at temperature T of thermal radiation of 
frequency/. 

Einstein tried to use Wien's original arguments plus the fundamental 
assumptions of the quantum theory in order to derive the Planck 
thermal radiation expression. In the course of his derivation l Einstein 
found it necessary to make certain assumptions about the fundamental 
processes involved in the absorption and emission of radiation. Be- 
cause the derivation was successful it must be assumed that these 
hypotheses were correct. 

Einstein's derivation is followed below in order to draw attention to 
these fundamental radiation processes and to establish certain quantita- 
tive relations between them. Both the fundamental laws and the 

1 "On the Quantum Theory of Radiation", Einstein, Phys. Zeits., Vol. 18, 1917, 
pp. 121-8. 



RADIATION AND MATTER-EINSTEIN'S RADIATION THEORY 



45 



quantitative relations are of great importance in the theory and design 
of masers and lasers. 

The discussion is essentially concerned with the energy associated 
with molecules, and two relations based on the quantum theory must at 
once be stated. 

First, only certain discrete states 1, 2, 3, ... n, arc permitted by the 
quantum theory and these will have energies E\, £2, £3 . . . £». 

Second, the relative frequency of occurrence (W n ) of the state n at 
temperature T is given by the statistical relation 



W n =p» **»'** 



• (B) 
(see also p. 3) 



where k is Boltzmann's constant and p n , the weight of the state, is 
independent of temperature and is characteristic of the nth state. 

To these two relations Einstein now added his three basic hypotheses 
about the radiation process. We shall consider these three hypotheses 
in relation to two states of the molecule characterised by energy £1 and 
£2, where E% is greater than £1, the molecule being able to pass from 
one state to the other by the emission or absorption of radiation at 
frequency/. 

(1) The probability that in time dt a molecule in the high energy state 
£ 2 will drop to the lower energy state £1 by the emission of energy 
£2 — £1 as radiation of frequency/ is given by 



dW = A u dt 



(C) 



where Az\ is a characteristic constant for the two levels considered and 
can be called the spontaneous emission coefficient. 

(2) In the presence of radiation of frequency / and density Ej the 
probability that a molecule in state E\ will in time dt absorb energy and 
rise to state £2 is given by 



dW = B12E/ dt 



(D) 



where B12 is a constant and can be called the absorption coefficient, or 
sometimes the induced absorption coefficient. 

(3) The converse process to (2). In the presence of radiation of fre- 
quency/and density £/ the probability that a molecule in state £2 will 
in time dt be induced to emit energy and fall to state £1 is given by 



dW = B 2 iE f dt 



(E) 



where B21 is a constant and can be called the induced emission co- 
efficient. 



46 



LOW NOISE ELECTRONICS 



These are Einstein's three elementary radiation processes and they 
are now applied to a collection of molecules at a temperature T. 

From the statistical expression (B) concerning the frequency of occur- 
rence of a given state 

we can write down the relative frequency of occurrence of the two states 
with energy E\ and £2 as 

p ier E t ikT anc | p2Z -E t tkT 

Now using (C), (D), and (E) above wc can obtain the number of each 
of the three radiation processes occurring in time dt. 
Number of spontaneous emission processes is 

p 2e -E,iicT ( ^ 21 ^ 

Number of induced emission processes in the presence of radiation of 
density Ef is 

p 2£ -E,lkT , £ f t Bzx dt 

Number of absorption processes in the presence of radiation of 
density Ef is 

pvL-BxlkT . E f . Biz dt 

If these radiation processes are not to destroy the equilibrium state 
then the number of emission and absorption processes must be equal. 

P2A2 1 b~e*I*t dt + p2B2iEfE-eJw dt = piBnEfz-Btim dt 
thus p2fr E * lkT [Aii + B 2 iE f ] */*«-* WBnEf . . (F) 

But the energy density of thermal radiation depends upon the 
temperature and if T becomes infinite then so does £/, 



i-e. P2[Azi + B 2 iEf] = piBi 2 E f 

ar| d />2#21 — Pi^l2 ■ 

Substituting this relation back into (F) gives 

p 2E -E t !kT[A 2l 4. BsiEf] ^p2B2lB-^ kT Ef 
£/P2^2l[e- £ i /A:r — t- B *l* T ] mm p 2 &- E * lkT A21 



■ (G) 



Ef¥ 



e -E t lkT 



A 2 i e - B ./* T — e-EJkT 



E f = 



_ A21 \Bn 



z E s -ElkT _ \ 






RADIATION AND MATTER— EINSTEIN'S RADIATION THEORY 



47 



This expression becomes identical with Planck's radiation distribution 
expression (p. 4) 



£y = W__J ) 

f C 3 \eW*T _ I) 



if £2 — Ei equals hf 
and 



A21 _ 871/f - a 
Bz~i~~c*~ J ' 



— g-/ 3 is sometimes written as ~-hf or Nulif, where Nm is the number 

of modes per unit volume (p. 3). 

Einstein thus succeeded in his attempt at an alternative derivation of 
Planck's expression and the hypotheses he adopted about the nature of 
the interaction of radiation and matter, (1), (2), (3), must therefore 
receive strong support. 

In developing his argument this far Einstein had considered only the 
energy involved in radiation processes, but now he extended it to include 
the momentum transfer taking place (cf, p. 37). 

". . .in general, we are content to limit ourselves to the considera- 
tion of the energy exchange, without taking into account the exchange 
of momentum. We can easily feel justified in so doing, since the 
smailness of the momentum transferred by the radiation makes this 
momentum insignificant in actuality in comparison with other causes 
of motion. But for the theoretical treatment, even this small effect 
must be considered just as important as the immediately obvious 
transfer of energy by radiation, because energy and momentum are 
so closely tied up with one another. Hence a theory can be regarded 
as valid only after it has been shown that the momentum which, 
according to this theory, is transferred from radiation to matter leads 
to the type of motion which the theory of heat demands." 

Einstein made the following assumptions about the transfer of 
momentum to a molecule when it is irradiated or when it spontaneously 
emits a quantum of energy. 

If a pencil of radiation acts so that a molecule encountered by it takes 
in or gives out a quantum of energy hf, then momentum hfjc will be 
transferred to the molecule — in the direction of propagation of the 
pencil if energy is absorbed by the molecule, and in the opposite 
direction if energy is emitted by the molecule. 

If a molecule gives out a quantum of energy by the spontaneous 
emission process, then it will be emitted in a random direction and the 



48 



LOW NOISE ELECTRONICS 



corresponding impulse suffered by the molecule will be oppositely 
directed. 

Using these assumptions he then showed that the motion thus 
imparted to the molecules would lead to exactly that distribution of 
velocities which was given by the kinetic theory of gases. 

Powerful additional support was thus given to the original hypo- 
theses, (1), (2), (3), and also to these additional ideas about momentum 
interchange. We shall now examine the significance of these ideas in 
maser and laser theory. 

If we are interested in amplification, then we must consider principally 
the effects (2) and (3) which occur when radiation is incident upon the 
material. If induced emission occurs, then the beam of radiation is 
enhanced, and if induced absorption occurs, then the beam is at- 
tenuated. Thus, for amplification we require as much stimulated 
emission and as little absorption as possible. 

Suppose that in the material there are Nz units in the higher energy 
state £2, and N% units in the lower energy state E\. 

Then the number of induced emission processes in unit time is 

NiBgiEf 

where B21 is the induced emission coefficient and £> is the density of the 
incident radiation. 
The number of absorption processes in unit time is 

NiB n Ef 

where Bn is the induced absorption coefficient. 

Each of these processes involves the emission or absorption of the 
same amount of energy £2 — £1 — hf, so that for amplification to occur 



i.e. 



But from (G) above 



N 2 B2iE f > NiBizEf 
N2B21 > N1B12 

P2B21 = P1B12 



and if we consider the simplest possible case where the difference in the 
statistical weighting factors pi and /> 2 can be neglected, then B 2 i = B&. 
The condition for amplification thus becomes Nz > N\ and the 
material must be maintained in a non-equilibrium state with more units 
in the higher than in the lower energy state. The normal equilibrium 
state is when #2 = # ir -{» t -*#*JP } so that energy must be supplied to 
the material in such a way that units in state Ei are pumped up to 
state £2. The non-equilibrium state when N<> > Ni is sometimes also 






RADIATION AND MATTER— EINSTEIN'S RADIATION THEORY 



49 



referred to as population inversion, or as negative temperature distribu- 
tion, and the achievement of such states is discussed in the next chapter. 

In the electronic processing of any information an important con- 
sideration is the signal to noise ratio. The noise which may be generated 
in an amplifier, and superimposed on the signal while it is passing 
through, is of especial importance if weak signals are to be handled. 

Spontaneous emission is a random process unrelated to the incident 
field and thus represents noise generated in the amplifier. 

It has already been shown that the ratio of the spontaneous emission 
coefficient to the stimulated emission coefficient is given by 



^21 
5 2 i 



Snhp 



Thus the noise generated in a stimulated emission amplifier will rise 
rapidly relative to the amplification produced as the signal frequency 
rises. Although such an amplifier used in the microwave region (maser) 
has an exceptionally good noise performance, the same would certainly 
not be true of a similar amplifier used at optical frequencies (laser). In 
fact, the laser is not likely to be generally used as an amplifier, certainly 
not of weak signals, but rather in the oscillator form as a generator of 
coherent light. 



NON-EQUILIBRIUM SYSTEMS 



51 



Non-Equilibrium Systems 



if the individual elements of any system can possess different amounts 
of energy, then there will normally be more with low energy than there 
are with high energy. A non-equilibrium system is one in which this 
normal state of affairs has been inverted, so that the population in a 
high energy level is greater than that in a lower energy level. 

Einstein has shown (p. 48) that in a two-level system the probability 
of a quantum of radiation being absorbed by a unit in the low energy 
level is equal to the probability of the radiation causing the emission of 
an extra quantum of similar radiation from a unit in the high energy 
level. If there are more units in the high level than in the low one, i.e. if 
inversion has been achieved, then there will be more emission than 
absorption and amplification will occur. 

In this chapter we shall examine the Einstein concept of stimulated 
emission, and the behaviour of the non-equilibrium system, in order to 
see how the two ideas may be brought together to achieve maser and 
laser action. 

Stimulated Emission 

The basis of maser and laser action is the stimulated emission of radia- 
tion, which was first recognised by Einstein in 1917 1 as one of the pro- 
cesses involved in the interaction of matter and radiation. 

He found a derivation of Planck's black body radiation expression 
which made use of the similarity in form between Planck's radiation 
distribution curve and Maxwell's curve of velocity distribution. The 
derivation was obtained by considering that the distribution of energy 
required by the quantum theory could only arise through the absorption 
and emission of radiation. 

Certain hypotheses about the absorption and emission of radiation 
by molecules formulated in the course of the derivation greatly clarified 

1 "Zur Quantenlheorie der Strahlung", Einstein, Phys. Zett., Vol. 18, 1917 (also 
p. 44). 






ideas about the interaction of radiation and matter. These ideas are 
conveniently considered in terms of a system of units with two per- 
mitted energy levels £2 > £1, such that transitions occur between them 
with the emission or absorption of a quantum of energy £2 — £1 = A/21. 
Three processes are possible in such a system: 

(i) A unit in the high energy level £2 may at any time emit radiation 
hfzi and fall to the state £1. This process will occur whether or not there 
is a beam of radiation of frequency /21 passing through the material. 
The quantum of radiation may be emitted in any direction and with any 
phase. The process is thus random and is called spontaneous emission. 

The second and third processes can only occur when a beam of 
radiation at frequency /21 is passing through the material. 

(ii) A unit in the low energy state £1 may absorb a quantum of 
radiation at frequency /21 from the beam and rise to state £2. In passing 
through the material the original beam of radiation will be attenuated 
by this absorption process. Algebraically, one may consider that 
photons of the same phase and direction as those in the original beam 
are subtracted from it. 

(iii) A unit in the high energy state £2 may be induced by an incident 
quantum to emit another identical quantum of radiation at frequency 
/21, in the same direction and with the same phase as the original 
quantum. The original beam of radiation will thus be amplified by this 
induced emission process. Algebraically, photons of the same phase 
and direction as those in the original beam are added to the beam and 
the induced emission process is the converse of the absorption process. 

There are two important quantitative results relating to these three 
processes which are used below. 

The first is that the absorption and induced emission probabilities are 
equal. Wn = W%\. 

The second is that the probabilities of spontaneous and induced 
emission are in the ratio («*/„/«• — 1) so that in the optical spectrum 
at room temperatures the ratio is very large, and spontaneous emission 
predominates, whereas in the microwave region stimulated emission 
predominates. 

Population Inversion — Non-equilibrium State 

If spontaneous emission is ignored, then the fact that the absorption 
and induced emission probabilities are the same {W\% = W21) means 
that amplification occurs when the population JVa of the higher energy 
state £2 is greater than the population Ni of the lower state £1. 



52 



LOW NOISE ELECTRONICS 



Such a state of population inversion was achieved in the first maser 2 
by making use of the two energy states of the ammonia molecule which 
are separated by an energy gap corresponding to a frequency of 
24 Gc/s. a An ammonia molecule in the higher energy state is repelled, 
and one in the lower energy state attracted, by a strong electrostatic 
field. So by passing a stream of molecules through a suitable electro- 
static separator it is possible to produce a gas with /V 2 P Nj. 



E 3 - 



SIGNAL 



N 3 



N 2 



PUMP 



■ Ni 



Fig. 6.1 

Physical separation of the high and the low energy units in this 
fashion is, however, not common and the principles of population in- 
version in the maser are best brought out by considering a three-level 
system operating in the microwave region. 4 

The choice of the microwave region allows two simplifications to be 
made in the theory. First, spontaneous emission may be neglected, and 
second, the linear approximation to the Boltzmann expression 



e -hfikT = i _ y,may be used, since at 3 Gc/s ^4x 10" 4 



kT 



Signal amplification is to be obtained at frequency fy> = 



£3 — £2 



and in order to do so a population inversion A/3 > iVg is required. This 
inversion is achieved by supplying separately a large amount of energy 
at frequency /31, the pump frequency, so that large numbers of units 
from state £1 are raised to state £"3. 

Under normal thermal equilibrium conditions before pump power is 



ipplied, AV= Jftr*f>»»t« M(l - ^)- 



Thus when the pump is first applied there will be more units which 
absorb energy than are induced to emit it. Upward transitions between 

2 "The Maser", Gordon et al., Phys. Rev., Vol. 95, 1954. 

8 "E-m Waves of 1-1 cm Wavelength and the Absorption Spectrum of Ammonia", 
Cleeton and Williams, Phys. Rev., Vol. 45, 1934. 

* "Proposal for New Type Solid State Maser", Bloembergen, Phys. Rev., Vol. 104. 
1956. 



NON-EQUILIBRIUM SYSTEMS 



53 



£1 and £3 will exceed downward transitions until eventually TVs would 
become equal to JVi. 

This assumes that transitions between the states are due only to the 
absorption or emission induced by the pump power. In the solid state, 
however, transitions between levels may be induced by interaction 
between the active units and the thermal energy (phonons) of the host 
lattice in which they are embedded. 5 Since the active units in the maser 
are usually paramagnetic atoms with the levels, £1, £2, £3, due to 
electron spin, this process is called spin-lattice interaction or relaxation. 

With practical maser materials the probability of spin-lattice inter- 
actions can be kept low (spin-lattice relaxation time high) so that near 
saturation pumping is, in fact, achieved and Ni and N3 tend to become 
equal when pump power is applied. 

If the spin-lattice transition probabilities are denoted by wis, M'31, 
H'i2» etc., then we can derive certain relations between them for use 
later by considering the state of thermal equilibrium when lattice 
induced transitions between states will be equal, i.e. ws&Ns = W&Nt, 
W21N2 = wnNi, etc. 

But in thermal equilibrium 



N 2 = Ni(\- h ^) and N 3 = nJ\ + 



hfa\ 
kT) 



using the Boltzmann approximation. 
Thus >vi2 = tV2i( 1 — jS J and 11-32 = Wtd 1 + -jS? ) 

When signal (jh) and pump (/13) power are applied, then, in addition 
to the above lattice induced transitions, radiation induced transitions 
occur. Let W23 W32 Wis W31 be the probability of such transitions. 

Then because induced absorption equals induced emission 

W23 = Wsz and Wis = Wsi 

Let us first consider a general case where radiation induced and 
lattice induced transitions may occur between all levels. 

—± = N*{WZ2. + w 32 ) + Ni(Wiz -f lt'12) — JV 2 (Wm + W<n + W 2 1 + M- 2 3> 

In the particular case we are considering, radiation is available only 
at frequency /32 (signal) and fsi (pump), so that W21 and Wn do not 
appear. 

Also from above W^3 = W32 and Wis = Wsi. 

5 "Masers", Weber, Rev. Mod. Phys., Vol. 31, No. 3, 1959. 



54 



LOW NOISE ELECTRONICS 



Thus ~-7~ = Wzz{Nz — N2) -{- (Wiwia — A^2W2i) + (A^j»v32 — JVaWas). 
But from above W12 = »W 1 — -j^ j and w^ = w&l 1 — 4M j 

N2hf32W32 _ Nif if nW2l 

kT 



kT 



dNt 



In the steady state —j- = 0, so approximately 

N3 - N , = *» / gBfe - W32/3. \ 
3kT\ FK32 + M'32 + W21 / 

where M + N 2 + N3 m N 

So that for population inversion between states £3 and £2, and con- 
sequently signal amplification at frequency /32, ^21/12 — W52/32 must be 
positive. The lower the temperature the more favourable the non- 
equilibrium state. 

If, as is often the case, the energy-level structure is roughly sym- 
metrical, so that/12 =?=/32, then there must be considerable disparity in 
W21 and h ! 32, the spin-lattice transition probabilities, if a reasonable 
degree of inversion is to be achieved. 

The signal power emitted by the maser will be (N3 — #2) W32AJ32 an0 " 
it can be shown that W32 increases if the interaction between the active 

units is small, i.e. if the spin-spin transition probability (—J is small, 

i.e. if t 2 the spin-spin relaxation time is high. 

The power output can, however, be seen to depend upon the con- 
centration of active units N. But as N increases so t 2 decreases, so that 
there will be an optimum concentration for maximum power output. 

In the first solid state maser, paramagnetic gadolinium ions were 
chosen as the active units because a convenient steady magnetic field 
gave an energy-level structure with transitions in the microwave region. 
The host lattice, chosen for its magnetic inertness, was lanthanum ethyl 
sulphate, and the concentration of gadolinium was 0-5%. 

The pump frequency was 17-5 Gc/s and the signal frequency 9 Gc/s, 
so that there was little disparity between /21 andyb. For good inversion 
it was therefore necessary that W21 and W32 should differ by as much as 
possible. It was found that doping the crystal with 0-2% cerium atoms 6 

6 "Use of Active Material in Three-Level Solid State Masers", SchuIz-DuBois 
et al„ Beii Syst. T.J., Vol. 38, 1959. 






NON-EQUILIBRIUM SYSTEMS 



55 



gave a ten to one disparity between the two spin-lattice transition 
probabilities and allowed maser action to be observed. 

Choice of Active Medium 

Having seen what is meant by population inversion and the influence 
upon it of transition probabilities, it is worth examining some other 
general principles which must be considered in choosing an active 
material for a maser or laser. To a certain extent the choice of material 
will be influenced by the convenience with which it can be pumped, but 
this aspect of the choice is deferred until the next section. 

The first consideration in choosing an active material is that it shall 
have an energy-level structure which permits transitions emitting 
energy of the frequency at which amplification or oscillation is required. 

Transitions at microwave frequencies can occur in free atoms and 
molecules, and some of these have been discussed in Chapter 2. Some 
of the gases in which such transitions take place occur naturally, like 
oxygen and water vapour in the atmosphere and like atomic hydrogen 
in interstellar space. Other gases like caesium and ammonia have been 
used in microwave devices like the atomic clock and the first maser. 
Conventional spectroscopy shows that there is a multitude of transitions 
in the optical range available in atoms in the gas and vapour states. 
The main problem in obtaining laser action is not the finding of a 
suitable energy-level system but the devising of a method of pumping. 
The noble gases, mercury, and water vapour are some of the materials 
in which laser action has been obtained. 7 

The energy-level schemes of atoms closely packed in the solid state 
becomes exceedingly complex because of interaction between the 
separate atomic systems (Chapter 3). In general, simple solid materials 
do not give line spectral transitions, although the presence of impurities 
may give absorption and emission at discrete frequencies. In particular, 
ions of the rare earth elements suitably dispersed as impurities in 
crystals, glasses, or even as organic compounds in solution, will allow 
the establishment of population inversion and show laser action. 8 If a 
suitable steady magnetic field is applied to certain single crystals doped 
with these rare earth ions, then microwave transitions and maser action 
can be produced, with the added advantage of tuning by varying the 
applied magnetic field. 

In any active medium there are conflicting requirements about the 
concentration of the units. A high concentration is required if the gain 

7 "Gaseous Optical Masers", Bennett, Applied Optics, Supplement 1, 1962. 

8 "Solid State Lasers", Davies and Moore, I.E.E. Laser Symp., September 1964. 

9 "Masers", Weber, Rev. Mod. Phys., Vol. 31, No. 3, July 1959. 



56 



LOW NOISE ELECTRONICS 



is to be high, while too high a concentration gives unwanted interaction 
between the units. In the gas devices, usually lasers, the concentration 
of active units is small so that the gain per unit length is also small and 
a very long resonant cavity, in fact, a Fabry-Perot system about a 
metre long, must be employed. Solid lasers and masers, very frequently 
ruby, have a higher concentration of active elements and with a higher 
gain can thus occupy a much smaller volume in either a cavity or a 
travelling wave system. It is, however, much more difficult to prepare 
the solid maser material as it must usually be cut from a perfect single 
crystal. Although the solid laser in transmitting systems can give very 
high pulse power, heating effects spoil the laser action and the mean 
power output is often no better than that of the gas laser, which is 
normally used if continuous operation is required, although CW 
operation of ruby lasers has been achieved. 10 

Pumping Methods 

In general for amplification to occur in a maser-type system an inversion 
of population must be achieved between the two levels which give a 
transition at the frequency to be amplified. Most practical microwave 
systems achieve this inversion by using a three-level system of the 
general type proposed by Bloembergen and discussed on p. 52. 



E* 



.M-. 



T 1 — 

SIGNAL 

i, N 2 



PUMP 



SIGNAL 




SPONTANEOUS 
EMISSION 



MICROWAVE 
E, I ^SIGNAL Nl 



(a) 



(b) 



Fig. 6.2 



Such a three-level system is shown in Fig. 6.2 (a) and, depending on 
the material chosen, it could operate either with signal due to transitions 
between levels 3 and 2, in which case population inversion between 3 
and 2 would be required, or with signal due to transition between levels 
2 and 1, in which case population inversion between 2 and 1 is required. 

In Fig. 6.2. (b) a similar three-level system is shown, but the gap 
between £" 2 and £ 3 is now very large and pumping from E\ to £" 3 is due 

'" "A continuously Operating Ruby Optical Maser", Nelson and Boyle, Applied 
Optics, Supplement /, 1962. 



NON-EQUILIBRIUM SYSTEMS 



57 



to optical energy supplied to the material. The microwave transition 
and the population inversion are between levels 2 and 1. 

This optical pumping of microwave masers shows promise of im- 
proving and extending their performance. 11 It enables the low noise 
performance of microwave amplifiers like the ruby maser to be realised 
at higher temperatures (e.g. liquid nitrogen) thus avoiding the necessity 
for very elaborate refrigeration. If a microwave pumped maser is 
operated at such a temperature the gain-bandwidth product and the 
noise performance are both poor, whereas optical pumping preserves 
the gain-bandwidth and noise performance and will also allow the 
maser to be operated at much higher frequencies. 



1 


W/////A 


/. ! 


E«_ 


™* P &SaL 
V 



ABSORPTION 
BAND 




ABSORPTION 
V BAND 

N 3 



TERMINAL 
j LEVEL Na 



(a) 



(b) 



Fig. 6.3 



In the case of lasers there are more pumping methods available than 
there are for microwave masers. We shall first discuss the optical 
pumping method which takes place in a fairly standard three- or four- 
level system in much the same way as has already been described. 

This optical pumping method is normally used in solid lasers. In gas 
lasers other more elaborate gas discharge pumping methods are avail- 
able involving collision between electrons and the gas, or between 
different atoms or molecules in gaseous mixtures; these methods are 
described later. 

The energy-level schemes used in the optical pumping of solid lasers 
are shown in Fig. 6.3. In each case the pumping is from the ground 
state Ei to a broad absorption band which is characteristic of the laser 
material, followed by a radiationless, phonon assisted transition from 
this broad band to a nearby sharp level Ez in (a) and £3 in (b). The 
laser transition then takes place between this sharp level and E\ the 
ground state in (a) or £2 the terminal level in (b). In each case a wide- 
band source of light such as a xenon flash tube or a mercury discharge is 
used as the pump. 

11 "Optical Pumping of Microwave Masers", Hsu and Tittel, P.I.E.E., Vol. 51, 
January 1963. 



58 



LOW NOISE ELECTRONICS 



In the three-level system (a), of which ruby is an example, a population 
increase is established between levels 2 and 1. A better system (b) 
establishes the inversion between level 3 and a terminal level 2 which is 
significantly higher than the ground state from which the pump is 
working. The pump thus has the large population of level 1 to pump 
from but only has to build level 3 up to a greater population than that 
of the terminal level 2. The population in level 2 will be very much less 
than that in 1 if the terminal level is a fair distance above the ground 
state. The establishment of inversion will occur for very much lower 
pump power, especially if the system is cooled, Calcium fluoride doped 
with uranium and cooled in liquid nitrogen uses a four-level system. 112 

In pumping a gas laser the fact that the uppermost energy level is not 
a broad band as in Fig. 6.3 but sharp like the other levels, means that 
only a very small fraction of the energy of a broad band pumping source 
is efficiently used. Optical pumping of a gas laser is only used if the 
frequency of the pumping transition required in the active material 
happens to coincide with the frequency of a strong line in the emission 
spectrum of some other gas. Under these circumstances quite efficient 
pumping can be achieved and such a system is used in the caesium laser 
pumped by light emitted from a helium discharge. 

The active element in a gas laser system is nearly always pumped 
into the high energy level by one of the many different sorts of collision 
process which may occur in a gas discharge. 13 This type of process can 
provide reasonably efficient pumping because a fairly wide band of 
incident particle energies can produce the required transition in the 
target atom or molecule, the excess energy being carried away as kinetic 
energy. 

In the simplest case electrons in the discharge excite the gas atoms to 
the required high energy level ; such a process with consequent popula- 
tion inversion may occur in a discharge in a pure noble gas. A second 
possible process involves a mixture of gases such as helium and neon. 
The population inversion occurs in the neon, but the original excitation 
is due to collision between electrons and helium and the subsequent 
transfer of the energy the helium has gained to the neon. There is a 
further set of processes where a molecule is given enough energy in a 
collision to dissociate into a pair of atoms, one of which is in the 
required excited state. 

The establishment of population inversion in a gas discharge is a very 
complex process with the possibility of several different contributing 
reactions in a particular case. Much research work is concentrated on 

12 Lasers, Lengyel, Wiley, 1962. 

13 "Gaseous Optical Masers", Bennett, Applied Optics, Supplement /, 1962. 



NON-EQUILIBRIUM SYSTEMS 



59 



understanding these processes and investigating new gases and mixtures 
in which inversion might be achieved. 

Other Considerations 

The general principles of non-equilibrium systems discussed in this 
chapter grew from Einstein's radiation studies (Chapter 5) and are 
themselves extended into the applied field in Appendices I and II. A 
quite different method of achieving a non-equilibrium system is dis- 
cussed in connection with the injection laser (Appendix V), and the 
phonon maser, 14 which is mainly of theoretical interest, uses an ultra- 
sonic wave to produce inversion. 

It is perhaps worth noticing that in this chapter it has been established 
that inversion can lead to amplification, but little has been said about 
the usefulness of such amplification. Applications are dealt with in the 
appendices, but it is appropriate in concluding this chapter to point out 
that spontaneous emission has not been discussed. The fact that 
spontaneous emission, and hence noise, increases so rapidly with fre- 
quency is responsible for the practical devices being split roughly into 
microwave amplifiers and optical oscillators. Oscillation, of course, 
implies the existence of amplification, but whereas the maser is a simple 
microwave amplifier invaluable for its low noise properties, the corre- 
sponding light amplifier would be of little use because of its very high 
noise figure. In the optical region the amplification principle is put to 
use indirectly to give oscillation and hence provide lasers, the high 
power coherent sources of light previously unobtainable. 

14 "The Phonon Maser", Tucker, Proceedings, Paris 1963 Quantum Electronics 
Conference, Columbia University Press, 1964. 



Power Relations in Parametric and 
Non-Equilibrium Systems 



in the maser amplifier we have already described a system where a 
signal at one frequency may be amplified if energy at another frequency 
is supplied to a suitable substance with the appropriate passive electrical 
circuits. Such amplifiers depend upon the upsetting of the normal 
equilibrium distribution of molecules among the various permitted 
energy levels of the active substance and are called non-equilibrium 
systems. 

In addition to the maser there is another class of amplifier which is 
supplied with energy at one frequency in order to amplify a signal at 
some other frequency. This other type of amplification is called para- 
metric and, like the maser, is important because of its low noise figure. 
The active element of a parametric amplifier is typically, but not always, 
a condenser whose capacitance can be made to vary with time. 

Although the parametric amplifier has been discussed theoretically 
for over a century and is essentially easier to make than the maser, it 
was the maser which was first produced as a working amplifier. 

Having already discussed the maser in some detail, we shall in this 
section state the most general properties of the parametric amplifier and 
derive certain important general principles which are common to both 
types. A discussion of practical parametric devices will be undertaken 
later. 

Types of Parametric Amplifier 

In the parametric amplifier, as in the maser, there are two frequencies 
introduced into the system from external sources. These are the signal 
frequency and the frequency of the pump which supplies power to the 
system in order that amplification may be achieved. 

The action of the time-varying parameter of the amplifier is to provide 
frequency mixing so that in the general case there will be generated the 



RELATIONS IN PARAMETRIC AND NON-EQUILIBRIUM SYSTEMS 61 

sum and the difference of the signal and the pump frequencies, and the 
sum and difference of all the harmonics of the signal and pump fre- 
quencies. The frequencies generated may be written as 

m = rt - — oo 

where m and n are integers, and o>i and w 2 are the original frequencies. 

In the practical parametric amplifiers of most interest the electric 
circuits are arranged so that only three of the multiplicity of possible 
frequencies can exist in the system. These three frequencies are wi, v^ 
and o>i + o)2 or mi — «2. Amplifiers which have power flowing at 
W1 _j- tog are called upper side-band amplifiers and those which have 
power flowing at wi — w 2 are called lower side-band amplifiers. 

Of the three frequencies present in the amplifier, one is that at which 
the signal enters (o,) and is normally fixed by factors outside the control 
of the designer of the amplifier. The second frequency, that of the 
pump (w P ), can be chosen by the designer quite freely, but once this is 
chosen then the third frequency can only have one of the two values 
w P ± w s . A suitable circuit to allow power to flow at the chosen fre- 
quency w p + cj s or w p — ta s must be incorporated in the amplifier. 

The frequency at which the low-power signal enters the amplifier is 
w fl and in many amplifiers this is the frequency at which the high- 
power amplified signal leaves the amplifier. Provided that power 
amplification occurs, there is often no objection to the signal leaving 
the amplifier at a frequency different from w s . In the case of the three- 
frequency amplifier considered above, four different arrangements 
appear to be possible : 

(i) Third or idler frequency w p — a 8 . Signal out at w«. 

(ii) Third or idler frequency « p — w 8 . Signal out at w P — w a . 

(iii) Third or idler frequency a p -f- &> 8 . Signal out at t* s - 

(iv) Third or idler frequency <a p -f g> s . Signal out at a> p + «*. 

In fact, only the arrangements (i), (ii), and (iv) give amplification and 
are used in practice. 

We require to find a set of general principles which will show us why 
arrangement (iii) is useless, and to give us some idea of the sort of per- 
formance we can expect from the useful arrangements (i), (ii), and (iv). 
The Manley-Rowe relations are a set of principles which provide this 
information and further guidance on the feasibility or otherwise of 
projected parametric systems of greater complexity, e.g. one allowing 
more than three frequencies to flow. 



62 



LOW NOISE ELECTRONICS 



Manley-Rowe Relations 

The Manley-Rowe relations 1 were originally derived for a system con- 
taining a non-linear reactance and a number of circuits tuned to 
particular frequencies. The results, however, are equally applicable to 
the maser, and Weiss 2 has pointed out that the derivation of the rela- 
tions is very much simpler in the quantum mechanical case. 

In order to see simply and quickly the form and usefulness of the 
Manley-Rowe relations we shall first consider the particular three- 
frequency system described above and shall use a quantum mechanical 
treatment to obtain the power absorbed or generated at the various 
frequencies. 

Power Relations in the Three-frequency System 

We shall consider an amplifier in which only three frequencies can exist. 
In a parametric amplifier this would mean that energy could only flow 
at three frequencies determined by the three tuned circuits contained in 
the system. Similarly, in the corresponding maser, energy could only 
flow at the three frequencies determined by the three permitted energy 
levels of the maser material. 



Wp-CO s 



Fig. 7.1 

Let two of the three frequencies in the system be co* and w p , the fre- 
quencies at which signal and pump power enter the amplifier. The 
third or idler frequency must now be chosen as « y — « g or w p -f w 5 . 

We shall first choose w P — a a when the three-frequency system in 
maser terms becomes as Fig. 7. 1 . 

£i, £2, and £"3 are the three energy levels, and under steady conditions 
the population of each level will remain constant. 

1 "Some General Properties of Non-Linear Elements", Manley and Rowe, 
P.I.R.E., Vol. 44, July 1956. 

2 "Quantum Derivation of Energy Relations Analogous to Those for Non-Linear 
Reactances**, Weiss, P.I.R.E., Vol. 45, July 1957. 



RELATIONS IN PARAMETRIC AND NON-EQUILIBRIUM SYSTEMS 



63 



There will be energy absorbed and liberated at all three frequencies, 
and each time a quantum of energy is absorbed or liberated one unit of 
the population will leave one level and go to another. 

Let Niz denote the nett number of units per second going from E\ to 
£2 with iVis and W23 referring to the nett number of transitions per 
second from £1 to £3 and £2 to £3 respectively. 

If the population in £2 is to remain constant 



N12 = N23 



(0 



But each transition from £1 to £2 means absorption of one quantum 
of energy £2 — £1 = hf s = sp^s- 
Similarly, each transition from £ 2 to £3 means absorption of energy 

~-( w » — 6> s ). 

Thus total power absorbed at signal frequency is 

Pa = Ni2^-^s 

Total power absorbed at idler frequency u p — w s is 



Pi ~ A^23t-(wj> 



Thus 



JV12 = -r • — and JV23 — ~£ 



Pi 



Wp — w s 



But from (i) above Nu = JV23 and thus 

Ps_ Pi 

Similarly, if the population of £3 is to remain constant, then 

Ms + ^23=0 . 



(A) 



(2) 



If we now consider the energy per quantum in transitions between 
£1 and £3 and bet wen £2 and £3 we shall obtain a relation involving the 
pump power. 

Power absorbed at pump frequency is 



Pp = Nl32-.Wp 



Thus Nn 



2nPx 



2tt Pt 



and we have already seen that Afo = -r 



ti>p — ag 



64 



LOW NOISE ELECTRONICS 



But from (2) above N& + N23 = 0, and thus 

Pi 



^ + 



(Up — Ct) s 



= 



But since Afa = ^23, a third relation can be written 

tlip tiig 



(B) 



• (Q 



Let us now collect together the three relations (A), (B), and (C) and 
examine some of their significance in the design and operation of 
practical parametric systems. 

It is well to remember that we are dealing with a particular type of 
amplifier in which the third frequency has been chosen to be (a p — a s ). 
This frequency will be referred to as «*, the idler frequency, as a con- 
venient shorthand. 

. . . (A) 



& + 3-0 

dip to,- 



^ + 



to 8 



= 



(B) 
(C) 



In our case energy is absorbed from the pump, which is the external 
power supply, so P P is positive. Thus from (C) P 8 must be negative, 
which means that more power is generated at the signal frequency than 
is absorbed from the signal source, i.e. there is power gain. 

Because the idler frequency is chosen to be o> ( = <a p — w a this is 
called a lower side-band amplifier, and because the amplified output is 
at the same frequency as the signal (w s ) it is called a straight amplifier. 



UJp 



CUp*ti) s =(i)i 



Fig. 7.2 



Suppose that we had decided to choose an idler frequency of ta v + o» 8 , 
this is then called an upper side-band amplifier and it can be represented 
as in Fig. 7.2. 



RELATIONS IN PARAMETRIC AND NON-EQUILIBRIUM SYSTEMS 65 
Using the same nomenclature as in (1) and (2) above 
JVi2 = JV23 and Nja + JV» = 

P 8 = Ma 5- "» 

J.TZ 



h_ 

2* 



Hence 



Pp = N23 tZ w p 



Ps Pp 

(&8 W J> 



(D) 



Thus 

Since 
and 



Pi = Nis o-( w j> + w «) = Ms «- «< 



Mf tl>p 



W12 = W23, Nib + N12, = 



(Hi «8 



(E) 



(F) 



Let us now collect the three relations (D), (E), (F), and see how this 
upper side-band amplifier can be made to provide gain. 



P* -Pv 



t» s 



tap 



* + 3> = 

o>f tig 



(D) 
(E) 
(F) 



In all amplifiers Pp is positive because the pump supplies power to 
the amplifier. From (D) P s must also be positive, which means that the 
amplifier absorbs more energy from the signal source than it liberates 
at the signal frequency. There is thus no possibility of power gain at 
the signal frequency and an upper side-band straight amplifier is not 
feasible. 

Power is absorbed at the pump and signal frequencies, but from (F) 

-K-P,? 

so that power is generated at the idler frequency. Since «* = w p + at 9 
Pi is greater in magnitude than P 3 and amplification of signal power is 



66 



LOW NOISE ELECTRONICS 



possible with the output now at a different and higher frequency than 
the input signal. This amplifier is accordingly called an upper side-band 
up-converter. 



The power gain ( G = ~ ) is — = ^ 
\ "s; w g 



+ <■»* 



= 1 H — -> and for a 

&>g <«J g 

given t» s a large value of pump frequency is chosen if large gain is 
required. 

Since P P = P s — > a large pump frequency implies that a high pump 

Wg 

power will also be needed. This is to be expected because high power 
gain means that the whole system is operating at a high-power level 
and will thus draw high power from its main energy source, the pump. 
We have seen that the upper side-band amplifier cannot be a straight 
amplifier but can give power gain as an up-converter where the power 
output is taken at the idler frequency. The lower side-band amplifier 
has already been shown to be capable of straight amplification and now 
examination of relations (A), (B), (C) will show that it can also give 
power gain with the output at the idler frequency. 



t')y, 0)( 



£0( 



Thus —Pi = — P p and power is developed at the idler frequency. 

Since taj = a p — ««, w* may be greater or less than u s and the amplifier 
may be an up- or a down-converter. 

The three possible arrangements to obtain power amplification in a 
three-frequency parametric amplifier may be summarised thus: 



Upper side-band amplifier 
Idler frequency <*n = w p + oig. 
Output at wf, hence up-converter. 

— Pt =— P s . Power gain—, which is always finite, so amplifier 
is stable. 

Lower sideband amplifier (i) 
Idler frequency to* = a> p — « g . 
Output at 6) S) hence straight amplifier. 

— P s — — Pp, Power output independent of power input at signal 

frequency, hence gain indeterminate. Unstable and may oscillate at 
signal frequency. 



RELATIONS IN PARAMETRIC AND NON-EQUILIBRIUM SYSTEMS 67 

Lower side-band amplifier (ii) 

Idler frequency «c = <o P — a s . 

Output taken at cu*. If to* > co s , amplifier is up-converter; if 
tot < cj s , it is down-converter. 

_i> ( = —P v . Power output independent of power input at signal 
ap 

frequency hence gain is indeterminate. Unstable and may oscillate at 
idler frequency. 

Practical Significance of Power Relations 

Let us consider in simplified equivalent circuit form the properties 
summarised above for the three amplifiers. 






Fig. 7.3 

In Fig. 7.3 the filters shown as square boxes limit the power flow in 
the three resistors R s , R Pi and Ri to the appropriate frequencies «g, 
oip, and cot. There are two generators in the system, the signal source 
outside the amplifier and the pump inside the amplifier. 

The resistor i? s , in fact, represents the internal resistance r of the 
signal source, the loss resistance r s of the signal circuit, and, in addition, 
Rt an effective load resistor if power is taken out at the signal frequency. 

If power is extracted from the idler circuit, then the load resistor 
appears as part of Ri, which now consists of the loss resistance n of the 
idler circuit plus ,Rl. 

Straight amplifier: R s = r + r s + Rl Rt = H 

Up- or down-converter: R 8 = r -\- r s Ri = n + Rl 

The power relations summarised for the three amplifiers in the pre- 
vious section hold for ideal conditions, e.g. lossless circuits. The 
inter-relation of r , r s , Rl, etc., are important in determining the per- 
formance of a practical amplifier. 

For example, the lower side-band amplifiers give output power, 
either at signal or at idler frequency, which is independent of the input 
signal power and there is thus a possibility of instability and oscillation 



68 



LOW NOISE ELECTRONICS 



at the output frequency which may be <o s or «* depending on the type 
of amplifier. 

The rest of the amplifier connected to the output circuit can be re- 
garded as a negative resistance because it is feeding power into the 
circuit. The magnitude of the negative resistance in the lower side-band 
amplifiers is controlled by the pump power. As long as the total re- 
sistance in the circuit has a nett positive value, then there is stable gain. 
If, however, in attempting to increase the gain, the total resistance is 
allowed to become zero or negative, then oscillation occurs. 



] AMPLIFIER 




INPUT 



3 OUTPUT 



Fig. 7.4 

In the straight amplifier the input signal and the amplified output 
both appear in the same circuit, so that the amplifier is a "one-port" 
device, i.e. the input and the output both use the same terminal. The 
difficulties of circuit organisation which arise with a one-port amplifier 
can be overcome conveniently by the use of a circulator, Fig. 7.4, at the 
sort of frequencies met in low noise applications (p. 1 27). 

Degeneracy 

In the lower side-band amplifier wt = a> p — w S) and it may be an up- 
or a down-converter depending on whether ^ is greater or less than w«. 

If (up is chosen to be equal to approximately 2« 8 , then u* and &>, will 
be different but nearly equal. 

In particular it can be arranged that the signal and idler frequencies 
are close enough together for both of them to lie within the bandwidth 
of a single amplifier circuit. The amplifier is now called degenerate. 

The degenerate parametric amplifier, needing only two circuits to be 
coupled to the active element instead of the usual three, is the simplest 
type to make. It also has the advantage of extra power gain because 
the total power at the signal frequency is nearly equal to the total power 
at the idler frequency. Since both lie within the same pass-band both 
may be detected and the effective gain doubled. 

The degenerate amplifier is, however, not invariably chosen because 



RELATIONS IN PARAMETRIC AND NON-EQUILIBRIUM SYSTEMS 69 

it has the inherent instability of the lower side-band amplifiers, and it 
may also not be the best choice in a particular application because of 
bandwidth and noise figure. 

Generalised Manley-Rowe Relations 

In the foregoing discussion the power relations quoted have been those 
relevant to the simple, practical three-frequency parametric or maser 
amplifier. They have been derived in quantum terms and not using the 
circuit approach used by Manley and Rowe in their derivation of the 
generalised expressions: 



I 2 

I I 

M=0 «{ - -oo 



mfi + nfa 



w , a. 



mfi + nfe 



z = 



In these expressions m and n are integers, so that the frequencies 
involved are/1,/2, and mfi + /1/2, where the sign may be positive or 
negative. 

The original paper (p. 62) or a detailed textbook 3 may be consulted 
for a full derivation which is of considerable mathematical complexity. 

3 Semiconductor Diode Parametric Amplifiers, Blackwell and Kotzebue, Prentice- 
Hall, 1961. 



8 

Parametric Diode and Ferrite Amplifiers 



in the previous two chapters we have discussed systems in which 
power at one frequency is amplified at the expense of power supplied 
to the system by a pump at some other frequency. Usually in such 
amplifiers there is power flow at three distinct frequencies. In the 
maser-type device these three frequencies are those of transitions 
between permitted energy levels and power is transferred between them 
by the mechanisms described in Chapter 6. 

In the parametric amplifier coupling between the two circuits carrying 
power at their separate resonant frequencies is carried out by a time- 
varying reactance pumped at a third frequency. The reactance may be a 
p~n junction acting as a capacitor and storing energy electrostatically, 
or a ferrite acting as an inductor and storing energy magnetically. 

The variable capacity parametric amplifier will be treated first and 
in more detail than the ferrite device because it is far more important 
practically, being in general commercial and military use. The ferrite 
parametric amplifier, although demonstrated in the laboratory, has 
been disappointing in practice. 

Variable Capacity Junction Diode 

The rectifying action of the junction between a />- and an n-type semi- 
conductor has been discussed earlier {Chapter 3). We arc here con- 
cerned particularly with the depiction layer formed at the junction. 
Positive and negative charge is stored in opposite sides of this depletion 
layer which has insulator properties, so that the p-n junction with its 
stored charge acts like a capacitor. We shall discuss the nature of this 
capacitor and show that the size of the capacitance depends on the 
voltage applied across the diode. If this voltage is made to vary with 
time, then the capacitance will also be time varying and can thus be 
used as the active element in a parametric amplifier. 

The depletion layer is that small region on either side of the junction 
which no longer contains free-charge carriers because electrons from 






PARAMETRIC DIODE AND FERRITE AMPLIFIERS 



71 



the n-type material have diffused into the />-type, and holes from the 
/7-type into the n-type material. The n-type material is left positively 
charged and the j5-type negatively charged by this diffusion process, 
which continues until a sufficiently great contact potential is established 
to prevent any further diffusion. It is possible to evaluate the potential 
difference across the depletion layer and the charge stored, thus ob- 
taining the capacitance. 

First let us examine the charge stored by the depletion layer. 

In the bulk of the material away from the depletion layer there is no 
nett charge because the free-charge carriers just neutralise the fixed 
impurity ions. Thus in the bulk of the n-type material there are free 



ABRUPT JUNCTION 
I 

IP 
l 
I 



GRADED JUNCTION 

IP 



I 4 

2 




Fig. 8.1 

electrons moving at random and a fixed pattern of positively charged 
impurity ions from which these free electrons originated. In the /?-type 
material the free carriers are positively charged and the fixed impurity 
ions are negative. 

If the free-charge carriers are removed, as they are in the depletion 
layer, then the charge of the fixed impurity ions is no longer neutralised. 
The exact charge density and distribution will depend upon the way in 
which the impurity atoms are arranged in the material. If the distribu- 
tion of impurity atoms in the neighbourhood of the junction is known, 
then integration over the thickness of the depletion layer will give the 
charge stored. 

The exact distribution of impurity atoms in the junction region 
depends upon the way in which the diode has been made, e.g. by 
alloying or by diffusion. In general, however, any diode will approxi- 
mate either to an abrupt junction of the type described in p. 26, or 
else to a graded junction where the concentration of impurity atoms 
changes approximately linearly with distance from/j-type to n-type. 

The resultant variation of charge density p with distance for a de- 
pletion layer of thickness d is shown in Fig. 8. 1 . 



72 



LOW NOISE ELECTRONICS 



Let us consider unit area of a graded junction where the charge 
density varies linearly with distance 

P = kx 

The total charge Q stored on each side of the junction is /p dx and 
thus 

Q oc*/ 2 

Now Poisson's equation relates electrical potential <jt and charge 
density 

ctv 2 e 

where i is the permittivity of the material. 

Integrating twice gives the total difference of potential over a de- 
pletion layer of thickness d as 



Thus 
and 
i.e. 



Vo oc d* 
Q oc d 2 
Vq ocd 3 



The ;„„,! chance isg ive„ by c = ^. 



The elastance S = 



I 



Thus C oc F -i and S oc K *. 

The relation for the abrupt junction calculated in a similar manner 
is C oc F<r*. 

Vo is the total potential difference across the depletion layer and 
consists of the contact difference of potential due to the hill in the 
energy-level diagram, plus any externally applied voltage. The exter- 
nally applied voltage may consist of a D.C. bias voltage plus any 
alternating voltage applied from the pump to cause capacity variation. 

We are normally only concerned with the time-varying portion of the 
capacitance caused by the alternating pump voltage at frequency v> p . 
For any type of junction the capacity may be written as C(r) indicating 
a time- varying capacity. 

C(t) = Co + Ci cos o p t + C2 cos Itupt -f Cz cos 3o> p f + . . . 

Different types of junction will have different values for the co- 
efficients. 
In all the previous discussion we have ignored any steady current 



PARAMETRIC DIODE AND FERRITE AMPLIFIERS 



73 



flowing through the diode which would constitute a condenser leakage 
current. The diode must be biased in such a way that such leakage 
current is very small and in fact the "built-in" bias due to the potential 
hill allows quite reasonable pumping voltages to be applied without any 
special external biasing arrangements. This matter is discussed more 
fully below. 

Another matter which has been ignored is the fact that the semi- 
conductor material on either side of the depletion layer will have a 
small but significant resistance. 

These effects can be accounted for by a leakage resistance across the 
condenser and a resistance in series with it, giving the simple equivalent 
circuit of Fig. 8,2 for the diode alone. 



>vw- 

— IK 



AW 



c 

Fig. 8.2 

In general, the leakage resistance is sufficiently high to be ignored in 
comparison with the condenser impedance at normal working fre- 
quencies. A more elaborate equivalent circuit including the effects of 
the diode container is discussed below. 



Impedance Matrix for Pumped Varactor 

The circuit of Fig. 8.2 strictly represents the varactor diode for a fixed 
applied voltage and hence a fixed value of C. If the diode is pumped at 
frequency up, then the capacity varies in the manner described in the 
last section. There will then be a complex relationship between the 
currents and voltages in the diode at various frequencies and the simple 
equivalent circuit will not help in evaluating the properties of any 
amplifier containing a pumped varactor diode. 

A useful tool in the analysis of parametric amplifiers is the impedance 
matrix representation of the varactor diode. The principles and method 
of application of the impedance matrix will be outlined below. 

The capacity of the pumped varactor has the form 

C(/) = Co 4* O cos t»pt + C2 cos 2« p f -j- . . . 
This may be expressed in the form 

C(t) = Co[l + 2yi cos apt + 2y2 cos 2<* p t + . . .] 



74 



LOW NOISE ELECTRONICS 



where yiCo is the half amplitude of the capacitance variation at t^p and 
yzC$ the half amplitude at 2oj>, etc. The half amplitude is used so that 
fractional indices are avoided when conversion of the cosine to the 
exponential form is carried out. It may be met in connection with any 
periodic quantity, e.g. voltage or current, 

2 cos = ei° + e-J° 

In the impedance matrix it is convenient to use the elastance rather 
than the capacitance. The relation between the time-varying voltage 
V{t) applied across the diode and the time- varying current f(t) flowing 

in it is then given by V(t) = R B I{t) -f fs(t)I(t) dt, where S(t) is the 

time- varying elastance of the pumped varactor and R s is the series 
resistance. 
Using the exponential form S(t) may be written 

S(t) = S + SieW + Si*e->M + $<&€&<*?*■ + S%*e~^^ + . . . 

where the coefficients St, S 2 , etc., refer to the elastance at the various 
harmonics of the pump frequency <*>$, and the asterisk indicates the 
conjugate complex. 

If we had been dealing with a very simple linear electrical component 
in which the current and the voltage were proportional, then we could 
write V = ZI, and we should call Z the impedance. In circuit analysis 
we should replace the electrical component by a box labelled Z which 
would tell us all we wanted to know about the current-voltage re- 
lationship of the component represented by the box. 

Suppose that we now have a more complicated electrical component 
in which currents and voltages at various different frequencies can 
exist at the same time. Let us start by considering a simple case in 
which only two such frequencies occur. A common relationship is 
then 

V\ = auh + #12/2 
V% = a%\h + 022/2 

where V\ and V2 are the voltages at the two frequencies, h and fa the 
currents, and the #'s are coefficients with the dimensions of impedance. 
Had there been n permitted frequencies, then there would have been 
n such equations and n 2 coefficients a. 



PARAMETRIC DIODE AND FERRITE AMPLIFIERS 



75 



A matrix is a convenient way of writing down such a set of n 
equations 






LVnl 



#11 #12 #1»' 

#21 #22 #2» 



,#«1 #m2> 



■ -#n«JL/«- 






or in the particular case of the two frequencies 



\V{\ = Van #i2ir/i1 



When the matrix relates currents and voltages in the manner shown 
above, then it is called an impedance matrix. The theory of matrices 1 
develops rapid methods of manipulating matrices in calculation, e.g. 
the inversion of an impedance to an admittance matrix or the simpli- 
fication of a large general matrix if certain terms are negligible. 

In particular, the previously quoted general equation relating signal 
currents and voltages in the pumped diode, 



V(t) = R s l(0 J rfs(t)I(t)dt 



gives rise to a many- term general-impedance matrix. But in the analysis 
of a particular circuit this general matrix will be considerably reduced 
by the recognition of certain simplifying features. Typically, a real 
problem might undergo gradually more detailed classification of the 
following type: 

(i) Small signals only are involved. The elastance variations are 
thus due only to the large amplitude pump voltages and are not 
affected by the signal voltages. 

(ii) A particular type of diode is used. The actual voltage-elastance 
law for this diode is now known, and certain of the elastance co- 
efficients, particularly perhaps those of higher pump harmonics, may 
be found to be negligible. 

(in) A particular amplifier may be under consideration, e.g. an 
upper side-band up-converter. Tuned circuits will now restrict the 
power flow to a limited number of frequencies. 



Determinants and Matrices, Aitken, Oliver and Boyd. 



76 



LOW NOISE ELECTRONICS 



The impedance matrix in a particular case is thus reduced from the 
multi-term general form to a simple expression like 

Si* 



ra- 



L j<*8 

which we shall apply below. 



J<*s 



JOJs 

J<*uA 



[a 



Application of Varactor Impedance Matrix 

If the reader is concerned with the calculation of the properties and 
performance of practical parametric diode circuits, then he should 
consult a textbook 2 which, although advanced in its field, treats this 
topic rigorously but from first principles in a manner similar to that 
used in the more familiar analysis of valve and transistor circuits. Only 
an indication of the form of such circuit analysis is given below. 

Suppose we consider the case of an upper side-band up-converter 
where the signal power enters at frequency t* B and the amplified power 
is extracted at a u = wj, + « 8 , where a p is the frequency at which the 
varactor diode is pumped. The incorporated tuned circuits restrict the 
flow of current to frequencies &> s and w u . The simplified varactor im- 
pedance matrix is given by 



ra- 



Rs + 



jv 3 









** + 



./««_ 



El 



(A) 



where V and / are the half amplitudes of the appropriate voltages and 
currents. This matrix is then used in the circuit given below (Fig. 8.3). 



r 



PUMPED 
VARACTOR 
IMPEDANCE 
MATRIX 



|f ^] 



Fio. 8.3 

In the circuit Zo is the internal impedance of the signal generator 
which provides an e.m.f. E, and Zl is the load impedance. 
Now from the circuit 

E - I s Zo + V s 
and this relation can be substituted into the matrix equation (A) to 
eliminate / s and V s . 

2 Varactor Applications, Penfield and Rafuse, M.I.T. Press. 



PARAMETRIC DIODE AND FERRITE AMPLIFIERS 



77 



In fact the matrix, which is really shorthand for two equations, is so 
simple that the reader might care to write down these two equations 
and then solve the circuit "longhand". 

The result is 



V* = 



Jto u 6> S 0) M 



1 



Zo -f R s + 



J<»$J 






Z + Rs + ~ 

jug 



This means that the amplifier can be represented by a generator 
giving an open-circuit e.m.f. equal to the second term, with an internal 
impedance equal to the term in the first bracket. 

The output impedance of the amplifier can now be written down and 
the output power readily obtained. A similar analysis gives the input 
impedance and, if noise voltages at the appropriate frequencies are 
added to the circuit, then its noise properties can be calculated. 

The various expressions obtained can be used to discover the best 
values for those circuit parameters which the designer considers to be 
especially important in a given application, e.g. there may be a certain 
load for maximum gain, and a different load for minimum noise figure. 
They also show how unavoidable features of the real circuits, like R s 
the series resistance of the varactor, reduce the amplification and noise 
performance to something less than the predictions of the Manley- 
Rowe relations which refer to perfect components. 



Practical Parametric Diode Circuits 

The foregoing discussion of the varactor diode parametric amplifier 
has been largely general and theoretical. This section mentions some 
more practical points like the biasing arrangements, the nature of the 
frequency-determining circuits, and a typical complete amplifier. 

In principle a p-n junction diode will act as a capacitor, and the 
reverse or forward current flowing will be small, for voltages from the 
breakdown voltage in the reverse direction to very nearly the contact 
potential, a fraction of a volt, in the forward direction. A steady bias 
voltage halfway between these two points allows the maximum applied 
pump voltage. If the diode is not to be fully pumped, then some other 
bias point may be chosen and in particular, in many cases zero external 
bias is used. The zero external bias condition is known as self bias 
because the rectifying action of the diode clamps the bias at a small 
negative value. 

The capsule and leads of a real varactor diode have distributed in- 
ductance and capacity which can reasonably be represented by L and C 



78 



LOW NOISE ELECTRONICS 



of Fig. 8.4. The diode thus has a natural resonant frequency and this 
affects the bandwidth of the amplifier. If the idler frequency is made 
equal to the diode resonant frequency, then the maximum bandwidth is 
obtained. 3 



-TRffiflP 






Fig. 8.4 

Parametric amplifiers are normally used at frequencies where the 
frequency selecting circuits can be produced by waveguide or coaxial 
line techniques rather than by using lumped capacitors and inductors. 
Such an amplifier operating in the region of 10 Gc/s might have the 
form shown in Fig. 8.5. 



INPUT -*-W s 




OUTPUT 



Fig. 8.5 



In the system shown the power is carried throughout by waveguides, 
the frequency of the wave in any particular guide being as indicated. 
The guide between the diode and the pump attenuator is cut off for 
a) a and &U and the pump attenuator controls the amount of pump 
power reaching the diode and thus acts as a gain control for the 
amplifier. 

The coupling is such that the signal circuit is loaded and hence 
broad-band while the idler circuit, unloaded except by the diode, 
largely determines the bandwidth of the amplifier which may be tuned 
by varying the pump frequency in step with the signal. 

The circulator converts the one-port parametric amplifier into a 
two-port device with input and output separated. The amplifier output 
is often to a mixer which is noisy and in such a case an extra circulator 
port terminated in a matched load is inserted between the output and 

3 " Varactor Diode Parametric Amplifiers", Hyde, Proc, I.E.E., Vol. 1 1 1, May 1964. 



PARAMETRIC DIODE AND FERRITE AMPLIFIERS 



79 



input ports to prevent mixer noise from being added to the original 
signal input to the amplifier. 

Improvements in bandwidth from about 10% to about 50% can be 
obtained by using a travelling wave parametric amplifier 4 which has 
about a dozen pumped varactor diodes acting as shunts in a waveguide 
or transmission line which carries both a signal and an idler wave. 
Such an amplifier has good noise performance and unilateral gain so 
that it may be used without a circulator. 

Performance and Applications 

Varactor diodes parametric amplifiers will operate up to about 100 Gc/s 
and have noise figures better than any other type of amplifier except the 
maser. Typical noise temperatures are a few hundred degrees absolute, but 
this can be improved to about ten or twenty degrees by the use of new junc- 
tion materials, like gallium arsenide, with liquid helium cooling. 5 

Except at frequencies above about 100 Gc/s, any low noise system 
can employ a suitable parametric amplifier 6 and current scientific 
papers and manufacturer's house journals continually refer to new 
applications in radar, terrestrial and satellite communications, and 
radio astronomy. 

Most operational parametric amplifiers are at present varactor diodes 
rather than any other sort (see below and Chapter 9), and they are 
particularly robust and do not necessarily demand refrigeration. 

An interesting application of the varactor diode in a non-amplifying 
role is in the generation of a high harmonic of a high frequency tran- 
sistor oscillator output so as to provide a microwave source which can 
replace the expensive small klystron, although the Gunn effect oscillator 
(p. 29) may prove an even better replacement. 

Other Solid State Parametric Amplifiers 

In principle any non-linear or time-varying reactance can serve as the 
coupling element in a parametric amplifier. In particular, an inductor 
might be used instead of the capacitor treated above. 

With the discovery of the ferrite materials there was a possibility of 
realising a low loss pumped inductor and a microwave parametric 
amplifier was proposed. 7 

4 "Investigation of an Experimental Travelling Wave Parametric Amplifier" 
Mavadoat and Hyde, Proc. I.E.E. (B), Vol. 109, September 1962. 

5 "Operational 4'2 3 K Parametric Amplifier", Stovman, P.I.E.E.E., Vol. 54, 
October, 1966 

6 Semiconductor Diode Parametric Amplifiers, Blackwell and Kotzebue, 
Prentice-Hall. 

7 "Proposal for a Ferromagnetic Amplifier in the Microwave Range", Suhi, 
Phys., Rev., Vol. 106, 1957, p. 384. 



80 



LOW NOISE ELECTRONICS 



The amplifier is pumped at *>„ and the other frequencies w s and «,- 
are present as in other parametric amplifiers, the active element in this 
case being a piece of ferrite with a steady magnetising field applied to it, 
usually in a cavity, although travelling wave versions exist. In general, 
though the ferrite amplifiers so far produced 8 have disappointed in 
efficiency and noise performance. 

More recently 9 a variable capacitor parametric amplifier has been 
reported based on the ferroelectric properties of a mixture of barium 
and strontium titanates. 

The complementary ferromagnetic and ferroelectric amplifiers now 
demonstrated in principle, but hardly comparable operationally with 
other parametric amplifiers, may well become more important prac- 
tically as new materials and more efficient methods of excitation become 
available. 

8 Coupled Mode and Parametric Electronics, LouiseiL Wiley, 
■ *'A Ferroelectric Microwave Parametric Oscillator", Pucel el ai., Prac, I.E.E.E., 
November 1963. 



Parametric Electron Beam Amplifiers 



in principle the pumping of any variable circuit parameter can lead to 
amplification. We have already discussed the variable capacitor and 
its practical realisation in the varactor diode. The variable inductor 
using ferrite has been made to work as a parametric amplifier, but we 
have seen that it has disappointed in practice because of difficulty with 
noise. There is an important and successful class of parametric 
amplifier where the pumped circuit parameter is not obviously identi- 
fiable as a condenser or inductance. This is the electron beam para- 
metric amplifier. 

When an A.C. signal is used to set up waves on an electron beam 1 
(Appendix III) it is recognised that the beam represents an impedance 
to the applied signal. Furthermore, this impedance can be caused to 
vary by the application of a pumping voltage to the beam in a suitable 
fashion. 

Parametric amplifiers have been constructed using electron beams 
carrying space charge waves of the type used in travelling wave tubes 
(p. 114). For various reasons, which are discussed later, the beam 
tubes using space charge waves were unsatisfactory, but a new type of 
beam wave, the cyclotron wave, was successfully employed. This wave 
is used in the Adler tube which is a low noise electron beam para- 
metric amplifier now common in communication systems. 

It is with the cyclotron wave and the Adler tube that we are princi- 
pally concerned in this chapter, although other types of amplifier are 
briefly mentioned later. 

Cyclotron Waves — Cuccia Coupler 

If an electron moves with a velocity v in a direction perpendicular to a 
magnetic field of strength B, then the electron will be acted upon by a 
force Bev perpendicular both to the magnetic field and its own direction 
of motion. The electron will move in a circular orbit with its plane 

1 The reader who is unfamiliar with electron beam amplifiers of the travelling wave 
type is advised to look at Appendix III before proceeding with this chapter. 



82 



LOW NOISE ELECTRONICS 



perpendicular to the magnetic field and with radius r, where 

mv 2 fr = Bev. The greater the electron velocity the greater the radius 

of the orbit. 

In the cyclotron the charged particle follows a circular path inside an 

evacuated enclosure consisting of two semicircular metal chambers 

between which an alternating voltage may be applied. There is a 

magnetic field perpendicular to the plane of these "dee"-shaped 

chambers. The voltage applied to the "dees" will give an electric field 

in the gap between them, which will accelerate or retard the charged 

particle each time it crosses the gap. If the alternating voltage has the 

correct frequency, then the particle will be accelerated each time it 

crosses the gap. The velocity will increase and the next semicircular 

orbit will be of greater radius, so that the particle will move in a sort of 

spiral path with continually increasing energy. The frequency of the 

Be 
alternating field for such cyclotron action to occur is given by « c = — 

In travelling wave tubes the beam is kept focused by a magnetic field 
which runs the length of the tube parallel to the electron beam velocity. 
Thus, if an electron in the beam is given a transverse velocity it will 
follow a circular path in the plane perpendicular to the beam velocity. 
The ordinary electron velocity down the tube will now have this per- 
pendicular circular motion superimposed on it and an individual 
electron will follow a spiral path. 

One way of giving an electron a transverse velocity is to apply a 
transverse electric field by allowing the beam to pass between a pair of 
deflecting plates. If the voltage applied to these plates alternates at the 
cyclotron frequency w c , then the electron will gain energy continuously 
while it is in the field, and the radius of the spiral it describes will in- 
crease as it passes through the plates. 

A pair of plates which impresses this cyclotron modulation on an 
electron beam is called a Cuccia coupler and the cyclotron frequency is 
in the hundreds of megacycles region for the sort of magnetic fields 
normally used for focusing in electron beam tubes. 

In Appendix III it is shown that longitudinal velocity modulation of 
an electron beam can give rise to a fast and a slow space charge wave. 
A similar investigation of the effect of transverse velocity modulation 2 
shows that four waves may be produced on an electron beam. 

These are the fast and slow cyclotron waves with velocities given by 



V = 



H 



8 Coupled Mode and Parametric Electronics, Louisell, Wiley. 



PARAMETRIC ELECTRON BEAM AMPLIFIERS 



83 



where » is the beam velocity, t» c the cyclotron frequency, and <» the 
frequency of the modulating signal. 

There are also two other waves, called synchronous waves, each 
having a phase velocity equal to the beam velocity. 

In practical amplifying devices it is the fast cyclotron wave with a 
forward (+ve) velocity which is used. In this case a snapshot of the 
beam taken at a particular instant would show it to be a spiral round 
the axis of the tube. At a later instant the pattern would have moved 
forward towards the collector end of the tube, its velocity being the fast 
wave velocity. Each individual electron has a forward velocity m and 
rotates about the axis at an angular frequency toe- 

The intersection of the beam with a plane perpendicular to it will 
describe a circle at angular frequency o». 

As the modulating frequency is allowed to fall to «o c the spiral opens 
out and becomes a straight line with infinite wavelength and velocity 
for the fast wave. We shall examine a little more closely what happens 
in the Cuccia coupler 3 which launches such a fast cyclotron wave with 
near infinite velocity. 

If power is to be fed on to a beam in the form of a wave, or removed 
from a beam where it exists as a wave, then the beam must be allowed 
to interact with a passive circuit which will support a wave of about the 
same velocity as the beam wave (p. 114). In the ordinary travelling 
wave tube, where power is to be exchanged with a slow beam wave, a 
suitable circuit which supports a slow wave, e.g. a helix, has to be used. 

Since the fast cyclotron wave is to be used in the particular amplifier 
we are now considering, then by choosing the signal and cyclotron fre- 
quencies to be approximately the same, the wave velocity becomes 
nearly infinite, and very simple lumped circuits can be used for coupling. 
Such a circuit, coupling only to the fast wave, is the Cuccia coupler, 
which is nothing more than two rectangular plates about a centimetre 
square and about a millimetre apart. All points along the plates change 
phase together so the circuit wave velocity is effectively infinite. The 
beam passes between these plates, and an alternating field applied to 
them gives the electrons transverse velocity and causes them to spiral at 
the cyclotron frequency. As they pass through the coupler, drawing 
energy from the alternating field, the radius of the spiral increases and 
then remains constant after leaving the coupler. 

If an electron beam already modulated with a fast cyclotron wave is 

passed through a Cuccia coupler then energy is extracted from the beam 

and delivered to a suitable load connected to the coupler. The electrons 

in the modulated beam will be moving in spirals of large radius when 

3 "The Electron Coupler", Cuccia, R.C.A. Review, June 1949. 



84 



LOW NOISE ELECTRONICS 



they enter the coupler, and this radius will decrease as the electrons lose 
energy in passing through the coupler. If the coupler is matched to the 
correct load, then all the fast wave energy is removed and the electrons 
no longer spiral when they emerge from the coupler. 

On p. 118 a reference is made to the "stripping" of noise from an 
electron beam. When a circuit wave is coupled to a fast beam wave 
there is an interchange of energy between them, the total energy of the 
system remaining constant. When an electron beam leaves the gun it 
will carry a certain amount of noise, some of which is fast wave noise 
which, in a fast wave device, would be added to the noise which was 
already present on the signal when it arrived at the amplifier input. 
But when this noisy beam interacts with the signal input circuit the 
signal power is transferred to the beam, but at the same time the fast 
wave noise on the beam is transferred to the circuit. 

If the input circuit is a properly matched Cuccia coupler, then all the 
available signal power is transferred to the beam as fast wave energy 
and all the fast wave noise is removed from the beam. 

Adler performed an experiment 4 in which a beam was passed through 
two Cuccia couplers which he referred to as lumped resonant cavities. 
The signal was applied to one of the couplers and the amount trans- 
ferred to a load connected to the other coupler was measured. At the 
same time the amount of noise appearing at the output was also 
measured. 

If the input circuit was tuned and the magnetic field was adjusted to a 
certain value, then there was a maximum transfer of signal power from 
input to output. The same adjustments also gave a minimum noise 
figure, thus substantiating the idea that the input circuit would couple 
signal into the beam and noise out of the beam. 

Adler suggested that a suitable device interposed between the input 
and the output couplers could be used to pump the beam and thus 
produce parametric amplification. In a later paper, 5 with others, he 
proposed that the quadrupole would be a suitable pumping device. 

The Quadrupole 

The quadrupole is an arrangement of four electrodes fed with A.C. 
power at the pump frequency in such a way as to provide a transverse 
circularly polarised electric field with radial components rotating about 
the axis at half the pump frequency, the field strength increasing with 

4 "Parametric Amplification of the Fast Electron Wave", Adler, P.I.R.E., Vol. 46, 
1958, p. 1300. 

5 "The Quadrupole Amplifier, a Low Noise Parametric Device", Adler et al., 
P.I.R.E., Vol. 47, 1959, p. 1713. 



PARAMETRIC ELECTRON BEAM AMPLIFIERS 



85 



distance off the axis. A similar field may be produced in a waveguide. 
Fig. 9.1 (a) shows the electrical connections from the pump to the 
four electrodes forming the quadrupole cavity. These electrodes should 
ideally be of hyperbolic section, but circular sections are also used. 
The plates are tuned with coils, and opposite plates are strapped 
together to ensure operation in a mode (ti mode) which gives the 
instantaneous polarities shown. 



X 


+ 

PUMP@ 



(a) 




Fig. 9.1 



In Fig. 9.1 (b) the plates are shown in thick section with instantaneous 
polarity marked and with equipotential lines shown dotted. The 
electron beam is passing through the quadrupole in the direction into 
the paper and the radius of the electron spiral at the section shown is 
indicated by the circle, the electrons rotating in a clockwise direction. 

The directions of the instantaneous forces on three electrons A> B, 
and C are shown by the arrows labelled F. Thus electron A will have its 
orbital velocity increased, C will have its orbital velocity decreased, and 
that of B will be unaffected because the force is perpendicular to the 
orbital velocity. 

The quadrupole field and the electrons are both rotating at the same 

frequency w c = -^> so that whatever condition {A, B, or C) the electron 

meets when it enters the quadrupole it continues to be accelerated, un- 
affected or retarded, as appropriate, all the time it is passing through 
the quadrupole field. When the electron is accelerated it gains orbital 
energy from the pump, its radius of gyration increases, and the fast 



86 



LOW NOISE ELECTRONICS 



PARAMETRIC ELECTRON BEAM AMPLIFIERS 



87 



cyclotron wave energy is increased, conversely the retarded electrons 
lose energy to the pump and the fast cyclotron wave energy is reduced. 

The field due to the quadrupole increases linearly with distance off 
the axis of the tube. If an electron is accelerated and has its radius of 
gyration increased when it enters the quadrupole region, then it will 
move in an orbit of greater radius, encounter a greater quadrupole field 
and thus have its radius of gyration increased still further. The radius 
on leaving the quadrupole is equal to the entry radius multiplied by an 
exponential factor which determines the gain of the tube and which is 
the same for large and for small amplitude signals, i.e. for large and 
small radius of gyration on entering the quadrupole region. The radius 
of gyration of electrons which are retarded will obey exactly the same 
law except that the exponential index will now have a negative sign. 

The gain is calculated by tracing the path of individual electrons as 
they travel through the quadrupole and then averaging all the possible 
conditions of phase. On average the exponential growth always exceeds 
the exponential decay and there is a resultant nett gain. 

Adler Tube or Quadrupole Amplifier 

The Adler tube using the fast cyclotron wave is the most important of 
the fast wave parametric amplifiers. Any parametric amplifier may be 
expected to have a good noise figure, but a simple electron beam 
usually contains so much noise that parametric amplifiers based upon 
electron beam tubes generally fail to achieve the expected low noise 
performance. 

The Adler tube has been a practical success for two fundamental 
reasons. Like any other fast beam wave device it is able to use the fact 
that fast wave noise can be removed from an electron beam by a suitable 
passive circuit before the signal is applied to it. But by using the 
cyclotron wave, which is essentially caused by electron velocities in 
directions perpendicular to the beam velocity, the Adler tube gets a 
further advantage, because transverse velocity noise is generated in an 
electron beam to a lesser extent than the longitudinal velocity noise 
which affects a space charge wave tube. 

There were really three stages in the development of the quadrupole 
amplifier or Adler tube. 

First, Cuccia invented the coupler which was a device for launching 
power from a generator on to an electron beam as a fast cyclotron 
wave, or for removing such power from the beam to a load. The 
electron coupler tube consisted essentially of an input coupler and an 
output coupler with an electron beam passing through them. The main 
purpose of Cuccia's tube was to act as a unidirectional connection 






between the power source connected to the input coupler and the load 
connected to the output coupler. It was also possible to modulate this 
power flow by varying the beam current. 

Second, Adler showed that the conditions for launching power from 
a signal source on to the beam in the input coupler were exactly those 
which would remove fast wave noise from the beam. Thus maximum 
power transfer from input to output was achieved with minimum noise 
figure. 



QUADRUPOLE 
INPUT OUTPUT 



COUPLER 



COUPLER 



GUN I 



x I ± 

T T T 



COLLECTOR 



SIGNAL 



PUMP OUTPUT 



MAGNETIC FIELD 



Fig. 9.2 

Third, Adler and others showed that a quadrupole device inserted 
between the input and output couplers could be used to pump the beam 
and amplify the power at the signal frequency appearing in the load. 

Figure 9.2 shows the layout of the important components of an 
Adler tube. The total tube length might be six or seven inches, with the 
axial magnetic field provided by a solenoid surrounding the whole tube. 
The signal input and output, and the pump input, would typically be 
through coaxial cables. 

The details of tube design and construction are covered in such 
references as 6, 7, 8, and only a few general remarks on design and 
performance will be given here. 

The electron gun will probably be of the multi-anode type with low 
noise temperature (p. 118) and this temperature may be further reduced 
by a high magnetic field in the vicinity of the cathode. The collector 
may be surrounded by electrodes designed to reduce secondary emission 
which causes extra noise, and the beam voltage will be quite low so that 
a reasonable number of electron orbits can take place in the length of 
couplers and quadrupole. With a magnetic field of 200 Gauss a beam 
voltage of 6 volts gives 4 orbits per cm. 

8 "An Electron-beam Parametric Amplifier for the 200 Mc/s Region", Chalk, 
Proc. IKE., Vol. 108, Part B, No. 37, January 1961. 

7 "A Microwave Adler Tube", Bridges and Ashkin, P.I.R.E., Vol. 48, March 1960, 

8 "Design of Cuccia Couplers for Quadrupole Amplifiers", Chalk, l.E.E. Elec- 
tronics Quarterly, December 1963. 



88 



LOW NOISE ELECTRONICS 



The axial magnetic field determines the value of a> e and must there- 
fore have the correct value to ensure that this is approximately equal to 
the signal frequency. The magnetic field is also necessary to keep the 
electron beam focused. Any interception of the beam by electrodes or 
other tube structure wilt cause an increase in noise level. Similarly, if 
large signals are being handled, the radius of the cyclotron orbits may 
become so large that collision with electrodes occurs with a similar 
increase in noise and with saturation effects on the amplifier gain, This 
last effect will limit the maximum power output of the tube. 

The quadrupole may be four curved plates or a cavity, and it is here 
that the amplification takes place, the power gain taking place entirely 
at the expense of power from the pump. The gain is independent of the 
frequency, i.e. the quadrupole itself has an infinite bandwidth. 

The Cuccia couplers may also be plates or cavities, the input and out- 
put couplers often being turned through 90° relative to each other in 
order to reduce electromagnetic coupling, and the bandwidth of the 
amplifier is determined by them. The efficiency with which they couple 
energy into and out of the beam affects the overall gain of the tube. 

The principal features of the performance of the Adler tube are that 
it is a low noise wide-band amplifier which is unconditionally stable and 
can be used at frequencies between a few hundred and a few thousand 
Mc/s. 

Fractional bandwidths of about 10% are obtained at a few hundred 
Mc/s and it has been shown 9 that this figure can be maintained into the 
microwave region if the current density in the beam is increased as the 
square of the frequency. 

Typically with present tubes the power gain might be 25 db, the noise 
figure less than 1 db, and the saturated power level about 20 mW. 

Other Electron Beam Parametric Amplifiers 

Although the Adler tube is the most important practical example, other 

members of the family of electron beam parametric amplifiers have been 

constructed. 

A transverse modulated electron beam in an axial magnetic field can 
support four waves (p. 82) and one of them is used in the Adler tube. 
Direct developments of this tube have been made using the syn- 
chronous waves, 10 and the slow and fast waves coupled. 11 This last has 

9 Seating Laws for Cyclotron-Wave Tubes, Bridges et al., Munich International 
Congress on Microwave Tubes, June 1960. 

10 "Amplification of Synchronous Waves", Lucken and Turner, P.I.E.E.E., Vol. 
51, September 1963. 

""The D.C. Pumped Quadrupole Amplifier— A Wave Analysis", Siegman, 
PJ.R.E.. Vol. 48, 1960. 












PARAMETRIC ELECTRON BEAM AMPLIFIERS 



89 



been the object of considerable interest because it operates on quasi- 
parametric principles using a D.C. "pump" which in this case is not 
the source of power for the amplification process. The noise per- 
formance has not, however, been as good as was at first hoped. 12 

There is no theoretical reason why parametric amplification of space 
charge waves should not be just as successful as parametric amplifica- 
tion of cyclotron waves. If space charge wave parametric amplification 
is to be attempted, then it is likely to be most profitable from the noise 
viewpoint if the fast space charge wave is used. This allows the tech- 
nique of fast wave noise stripping described before to be carried out. 

A number of such parametric amplifiers have been described, 13 - 14 but 
the performance has not so far been as good as that achieved with 
cyclotron waves. 

12 Experiments on the Noise Performance of a D.C. Pumped Quadrupole Amplifier, 
Vokes and Bridges, NATO-Agard Conference on Low Noise Electronics, Pergamon, 
1962. . 

13 *' Parametric Amplification of Space Charge Waves", Ashkin, J. App. Phys., Vol. 
29, December 1958. 

14 "Some Notes on the History of Parametric Transducers", Mumford, PJ.R.E., 
Vol. 48, May 1960. 



Appendix I 

Practical Maser Systems 



the fundamentals of maser action and design have been discussed 
elsewhere, but little has been said of the way in which this particular 
type of low noise amplifier is fitted into a microwave system. New 
features like circulators and refrigeration are needed in maser systems, 
while some existing components, like duplexers and even waveguide 
feeders, require to be redesigned and improved. 

In the years immediately after the first successful operation of the 
three-level solid-state maser these amplifiers were incorporated into 
existing radio systems while still in an experimental state. Now, how- 
ever, systems are appearing which have been designed with the maser as 
an integral part. In this section the "hardware" associated with the 
maser, and some typical special features of such new communication 
and radar systems are outlined. 

Maser Structure, Magnet, Pump 

The maser structure itself and the associated magnet and pump are the 
most obvious novel features of any system, together with the refrigera- 
tion which is described later. 

The active maser material is usually ruby, which must be inserted 
into a cavity, or into a waveguide for a travelling wave maser, with 
circuit arrangements so that pump power can be provided to the ruby 
and signal power taken to and from it. 

A very simple cavity for an X-band maser pumped at K-band, con- 
sists of a piece of ruby of the correct size (0-68 x 0-5 x 0-45 in.) with 
its surface silvered. 1 Two slots cut in the silver layer provide the micro- 
wave coupling at signal and pump frequencies and such a cavity can be 
clamped or soldered to the feeder system. 

Two typical feeder arrangements from modern systems 2 - 3 are shown 

1 "Silvered Ruby Maser Cavity", Cross, J.A.P., Vol. 30, September 1959. 

2 Schimitschek et al„ P.I.E.E.E., Vol. 51, February 1963, p. 363. 

3 "Masers in Mars Radar Expt.*\ Higa and Clauss, PJ.E.E.E,, Vol. 51, June 1963. 









PRACTICAL MASER SYSTEMS 



91 



in Fig. 1.1 The system shown in (6) employs a coaxial signal feed rather 
than a waveguide and was one of two masers used in series in the Mars 
radar experiment. 



PLATED RUBY COUPLING HOLES 



SHORT 




(a) 



; 



PUMP 
W.G. 



lSC 



^^ 



TUNING RUBY 

SCREW 



(b> 



.SIGNAL 
COAXIAL 



Fig. 1.1 

In the travelling wave maser, which in general has better bandwidth 
and power handling capacity than the cavity maser, the ruby material is 
contained in a waveguide. 

A comb structure made of a number of posts (Fig. 1.2) slows down the 
signal wave in the guide so that it is in contact with the active material 



RUBY- 



i 



-POST 



,FERRITE 



Fig. 1.2 

for a reasonable time. The pump power is propagated down the guide 
as a fast wave. In the section illustrated, the ruby is mounted on one 
side of the post, while the ferrite shown absorbs any backward travelling 
wave and improves the stability. 

A typical travelling wave structure, that used in the Telstar com- 
munication system, 4 was 5 in. long, and the comb contained 62 fingers 
loaded on both sides with ruby. 

The magnetic field must be at a given angle to the crystal axis of the 
ruby and its magnitude determines the frequency of operation. The 
field in the typical travelling wave maser quoted above was 0-33 Wb/m 2 

4 "Masers for Telstar Satellite", Tabor and Sibilia, Bell Syst. Tech. J., Vol. XL1I, 
No. 4, Pt. 3, July 1963. 



92 



LOW NOISE ELECTRONICS 



provided by a permanent magnet which was kept at a constant tempera- 
ture because the field variation was 4 x 10~ 5 Wb/m 2 per degree Centi- 
grade, and the tuning variation was 2400 Mc/s per Wb/m 2 . 

The pump was a reflex klystron with an AFC system which kept its 
frequency within a few Mc/s of 30,180 Mc/s. It delivered about 
70 mW of power since a fall in pump power below 50 mW would reduce 
the maser gain by 2 db. 

Noise Temperature Due to System Elements 

Circuit elements which arc intrinsically noisy like mixers, and lossy 
components like feeders and duplexers, all increase the total noise 
temperature of the system. The reduction of such effects may require a 
new fundamental principle in the system design, or merely piecemeal 
improvement in the individual components. 



aeU- 



DUPLEXER 



MASER 



MIXER 



EQUALISER 



I.E 



TX 



Fig. 1.3 



As an example of the low noise requirement radically affecting the 
system layout, the R.F. stage of the Telstar receiver system is worth 
considering. 

The chosen maser at liquid helium temperature had 42 db gain with 
16 Mc/s bandwidth between 3 db points, while at 25 Mc/s bandwidth 
the gain had fallen to 35 db. The system was, however, required to 
have a bandwidth between 3 db points of 25 Mc/s, and this could have 
been achieved by stagger-tuning the maser. 

The operating frequency of any maser is determined by the applied 
magnetic field. If the first half of the structure of a travelling wave 
maser is in a different field from the second half, then the two halves 
will tune to different frequencies and staggered tuning will result. A 
bandwidth of 25 Mc/s can be obtained but the gain is only 27 db, which 
would be adequate if it were not for the very large noise temperature of 
the mixer stage which follows the maser. 

This mixer has a 12 db noise figure which corresponds to a noise 
temperature of (15-9 - 1)290 = 4321° K. 

If the maser gain is 27 db (500)— the stagger-tuned value— then the 
contribution of the mixer to the noise temperature at the maser input 
terminals is 4321/500 = 8-6° K. This would greatly degrade the noise 









PRACTICAL MASER SYSTEMS 



93 



temperature at the maser input because the figure due to the maser and 
its input waveguide alone is only 3-5° K. 

With the single-tuned maser the gain is 42 db (15,894) at mid-band, 
and 35 db (3162) at 25 Mc/s bandwidth. 

Thus the noise temperature contributed by the mixer at the maser 
input terminals is 4321/15,894 m 0-27° K at mid-band, and 4321/3162 
= 1-36° K at the band edge. 

The receiver response was now given the required shape by the 
equaliser circuit, a bridged T filter inserted after the mixer, which gave a 
7 db loss at mid-band and 3 db loss at 25 Mc/s bandwidth. The overall 
gain thus became 42 — 7 = 35 db at mid -band and 35 — 3 = 32 db at 
the band edge. 

In contrast to the foregoing example of a fundamental modification 
to system layout consequent on the use of maser amplification, the first 
successful use of a maser in a conventional X-band radar, of peak pulse 
power 150 kW, necessitated the improvement of one of the com- 
ponents — the duplexing system. 5 

The conventional TR switch allowed a few milliwatts of transmitter 
power to break through to the maser. This reduced the population in 
the higher level and caused the maser gain to fall. In addition, the 
breakthrough power varied from pulse to pulse and the fluctuation in 
receiver performance was unacceptable. An extra 30 db of isolation 
was required to protect the maser from saturation. 



AE 



T.R. 




FERRITE 




MASER 




MIXER 




LK 


SWITCH 


SWITCH 








i\ 




A 








TX 


SYNC 


CURRENT 
DRIVING 










PULSE 








Fig. 1.4 

A high-speed ferrite switch, inserted between the TR switch and the 
maser, provided 30 db of isolation during transmission, but only 
0-25 db of loss, corresponding to 0-06 X 290 * 17° K increase in the 
system noise temperature. 

The switch consisted of a tapered ferrite rod in the centre of a wave- 
guide with conducting fins on each side of the rod. With no D.C. 

5 Goodwin, P.I.R.E., Vol. 48, January 1960, p. 113. 



94 



LOW NOISE ELECTRONICS 



magnetic field applied the ferrite acts as a dielectric and the fins are in 
an equipotential plane and cause no attenuation of the normal mode. 
During the transmitted pulses (2-35 microseconds long at 416 p.r.f.) a 
current pulse produces a longitudinal magnetic field which is applied to 
the ferrite. The wave is no longer propagated unaffected through the 
structure (Appendix IV) and more than 30 db of attenuation is ob- 
tained over a bandwidth of 120 Mc/s. 

The noise temperature of the whole system was 173° K, which com- 
pared well with the figure of about 2000° K for a good X-band radar 
with no maser. 

Refrigeration 

Perhaps the greatest novelty for the electronic engineer in a low noise 
microwave system is the refrigeration, although those who have worked 
with infra-red detectors will already have encountered it. 

In general, liquefied nitrogen (77° K) or helium (4-2° K) in a dewar 
vessel provides the low temperature environment. Because of its 
cheapness compared with helium, nitrogen is used when possible, either 
as the only cooling if the system does not demand the very lowest 
temperatures, or as a precooling liquid in a double dewar or some other 
arrangement, to reduce the loss of liquid helium which is used to provide 
the final cooling. 

In a maser system the low temperature is provided for two reasons: 
first the maser will not work at all unless its temperature is well below 
room temperature (p. 57), and second, the degradation in noise per- 
formance due to lossy elements can be much reduced if they are cooled. 

Typically an element like a circulator occurring in a system before 
the first amplifying stage will, if lossy, increase the noise temperature of 
the system considerably. An element at room temperature with 0-5 db 
loss has an absorption coefficient of 0TI and will increase the noise 
temperature by about 32° K, while at helium temperature the degrada- 
tion will be less than a degree. 

It is therefore worth while to design a helium-cooled circulator to 
operate with a travelling wave maser. 6 One of the main considerations 
in this design is thermal performance so that loss of liquid helium is not 
too great. This particular circulator dissipated 35 mW of heat, made up 
of joule heating in the coils generating the magnetic field for the 
ferrite circulator, and eddy current losses due to switching of this field. 

Quite frequently, where the introduction of a maser has necessitated 
helium cooling, thermodynamics plays an important part in the system 
performance. 

8 de Gray] et a/., P.I.E.E.E., Vol. 51, June 1963, p. 947. 












PRACTICAL MASER SYSTEMS 



95 



For instance, a travelling wave maser may be able to tolerate, without 
disturbance to its output, one or two watts of power outside its pass 
band breaking through to its input terminals from a neighbouring 
transmitter. But the dissipation of so much power may involve an 
unacceptably large loss of liquid helium, so that better isolation re- 
ducing the total breakthrough at all frequencies to not more than a few 
hundred milliwatts is required. The isolation at the signal frequency 
must, of course, be much better in order to avoid saturation. 

The maser input leads have always made the largest contribution to 
the noise temperature of the maser amplifier itself and their design 
involves conflicting electrical and thermal requirements. Thin wall, 
stainless steel coaxial cables reduce heat conduction, and thin copper 
plating improves electrical properties, a compromise being reached 
between the relative amounts of steel and copper employed. 

In the Tel star maser (p. 92) waveguide input leads were used instead 
of coaxial to get the lowest losses. The waveguide was 0-020 in. thick 
seamless stainless steel internally plated with 0-0002 in. copper. The 
room temperature loss of the maser feed with this guide was about 
0-1 db. A number of such guides were made and selected for the best 
noise performance. 

Some typical performance figures of the Telstar system are of interest 
in terms of the application problems discussed above: 



Centre frequency 

Effective bandwidth . 

Effective gain . 

Pump frequency 

Pump power . 

Magnetic field . 

Overall maser noise temperature 

Bath temperature 

Liquid helium consumption 

Helium capacity 



34-5 



4170 Mc/s 

25 Mc/s 

db (equalised) 

30,175 Mc/s 

. 70 mW 

0-33 Wb/m 2 

. 3-5° K 

. 4-2° K 

i litre/hour 

10 litres 



Future Development 

It seems likely that future development in maser systems may lie in 
three main fields: the maser itself, the associated radio equipment, and 
the cryogenic system. 

In the maser itself materials, pumping, and operating techniques are 
likely to be examined with a view to improving the gain-bandwidth 
product at a given temperature, raising the operating temperature, 



96 



LOW NOISE ELECTRONICS 



improving the efficiency, and extending the frequency range particularly 
in the higher direction. 

Optical pumping in a ruby maser was first demonstrated some years 
ago 7 and more recent work 8 - 9 has shown that optically pumped micro- 
wave masers can be operated at much above 4° K, typically at liquid 
nitrogen temperature and higher, without the deterioration in noise 
temperature and gain-bandwidth product which occurs with microwave 
pumping at elevated temperatures. In addition, the use of optical 
pumping allows the maser frequency to be extended beyond the range 
of present devices using microwave pumping sources. 

We have already discussed some of the ways of reducing the de- 
gradation in noise temperature due to the associated circuit elements 
like feeders, duplexers, switches, etc. The figures quoted below on the 
Telstar system noise temperature, together with those quoted earlier 
(p. 9) on the Echo project, give some idea of where there is scope for 
improvement, and of those contributions like the 2-4° K due to the 
atmosphere, which cannot be avoided — except by going outside the 
atmosphere into space. 

The contributions to the total noise temperature with the aerial at 
90° elevation are: 



Absorption in dry atmosphere 
Aerial side lobes . 1-0" 
Feeder circuits, etc. 14-2 
Maser . . . 3*5 
Second stage . 0-5 

Radome absorption (dry) 

Radome scattering (dry) (ground noise scattered into 
aerial horn) 



2-4° K 
19-2° K 

3°K 

7-4° K 
32° K 



The noise temperature increases to 42° K at 7*5° elevation due to 
greater path in the absorbing atmosphere. It rises very sharply to over 
200° K when the aerial beam strikes the ground. A wet atmosphere 
also causes an increase, and a wet radome may put the temperature up 
by over 20° K. 

Perhaps most progress is likely to occur in the near future in the 
cryogenic techniques. Closed-circuit helium liquefiers are now being 

' Devor et al., Phys. Rev., Letters 3 r November 1959, p. 468. 

8 Hsu and Tittel, P.I.E.E.E., Vol. 51, January 1963, p. 185. 

9 Szabo, PJ.E.E.E, Vol. 51, July 1963, p. 1037. 



PRACTICAL MASER SYSTEMS 



97 



developed and some thousands of hours experience have been acquired 
in operating masers in such systems. 10 Typically, a system will consist 
of a compressor and a refrigerator unit which may be separated by as 
much as 200 ft. Ten days of operation of the closed-circuit system with- 
out adjustment has been reported with the refrigerator unit mounted 
on the aerial of a low noise system. 

» Higa, Wiebe, P.I.E.E.E., Vol. 51, May 1963, p. 851. 



LASER SYSTEMS AND APPLICATIONS 



99 



Appendix II 

Laser Systems and Applications 



the general requirements for the achievement of a non-equilibrium 
system at optical frequencies have been discussed elsewhere (p. 48 and 
p. 57). The purpose of this appendix is to describe some typical laser 
systems with their associated equipment and to discuss some of the 
applications of the laser. 

The laser is almost always met as a radiation source, i.e. as an oscil- 
lator rather than as an amplifier. The amplifier is therefore not dis- 
cussed until the end of this appendix and we must start by considering 
how an active laser material can be made into an oscillator. 



FULLY 
REFLECTING 



V 



PARTIALLY 
REFLECTING 



ACTIVE GAS 



/ 



J 



FULLY 
REFLECTING 



--^OUTPUT 



R.F. ELECTRODE 



(i) 




Fig. II.l 

The general requirements in any oscillator are power gain, provided 
in the laser by the active material, feedback of sufficient power to main- 
tain oscillation, and a method of radiating the remainder of the power 
externally. We shall first consider the sort of system which provides the 
required feedback and output. 

In general, the active laser material is mounted between two parallel 
mirrors as shown in Fig. II.l (i) and (ii). In the case of the gas laser 1 the 
mirrors form the ends of the tube in which the discharge is maintained 
by an external coil or coils connected to a radio frequency oscillator. 
In the solid laser the active material, e.g. ruby, in the form of a rod has 
optical flats ground on the ends and these are coated to give the required 
reflection. The ruby is pumped to the active state by the light output 
1 "Gas Lasers", Bloom. P.I.E.E.E., Vol. 54, October 1966. 






from a flash tube, containing for example xenon, which is coiled round 

the rod. 

One of the mirrors is made totally reflecting and the other partially 
reflecting, so that enough energy is fed back through the active material 
to maintain oscillation. The light escaping through the partially re- 
flecting mirror is the laser output. The space between the mirrors is, in 
fact, a resonant cavity many wavelengths long, whereas in the micro- 
wave maser the active material is often mounted in a cavity with di- 
mensions of the order of only one wavelength. The whole of the cross- 
section of the partially reflecting mirror constitutes the source of light, 
and it should be filled with radiation of one frequency, phase, and 
intensity. This is generally the case with a gas laser, but the solid laser 
tends to give a number of spots on the end of the rod rather than 
uniform illumination. The light from different spots will be of different 
frequency, phase, and intensity, each spot representing the output due 
to one particular mode of oscillation of the long optical resonator. 
Inhomogeneities in the crystal encourage the occurrence of filamentary 
laser action along separate paths in the rod, and since the conditions 
for oscillation vary slightly with temperature, the general character of 
the laser output tends to change during the time of operation. Much 
of the energy provided by the flash tube tends to heat the ruby and the 
dissipation of this heat is a difficulty. Consequently most, but not all, 2 
solid lasers operate under pulsed conditions, so that the temperature 
rise is not too great. Furthermore, the flash tube is required to produce 
several hundred joules of pumping energy in order to maintain laser 
action, and this is normally obtained by discharging a bank of con- 
densers through the lamp under pulsed conditions. Less than 1% of the 
energy supplied to the ruby appears as coherent light, and without 
elaborate precautions 3 there is a tendency for the output to be spiky 
rather than a clean, square pulse of light. 

Both the gas and the solid laser show this very low efficiency whereas 
that of the injection laser (p. 129) is very much greater. In general, the 
gas laser provides better coherence and easy continuous wave action, 
while the solid laser can provide very high peak power output of the 
order of megawatts. 

Modified Laser Oscillator Systems 

The plane parallel mirrors placed at either end of the active laser 
medium constitute a Fabry-Perot interferometer system. It has been 

2 "A Continuously Operating Ruby Optical Maser", Nelson and Boyle, Applied 
Optics Supplement J, Optical Maser s, 1962. 

3 Lasers, Lengyel, Wiley. 



100 



LOW NOISE ELECTRONICS 



shown 4 that much greater separation of the possible resonator modes 
will occur, and much less power loss due to diffraction of the parallel 
beam at the mirrors, if a confocal mirror system is employed instead of 
the plane parallel arrangement. Two spherical mirrors are set up so 
that the centre of curvature of each lies in the surface of the other and 
their foci coincide in the centre of the active material. These mirrors 
are now outside and quite separate from the ruby rod or the gas dis- 
charge tube. In the latter case a considerable advantage is gained 
because the comparatively delicate highly reflecting mirror coatings no 
longer suffer the baking or other outgassing to which the discharge tube 
must be subjected. One of the principal advantages of the confocal 
mirror system is that the angular tolerances for oscillation to occur are 
much less stringent than in the plane parallel case and the laser is very 
much easier to set up. 

When the plane mirrors are removed from the ends of the discharge 
tube in the confocal system then the tube must be closed by windows 
which contain the gas but allow the multiple passage of a light beam 
with the minimum loss due to absorption or reflection. This is com- 
monly done by sealing the ends of the tube with optical flats made of a 
suitable low absorption glass and set at the Brewster angle to minimise 
reflection. 

The problem of getting the laser light out of the gas laser has its 
complement in the problem of getting the maximum amount of pump 
light into the ruby. A composite arrangement using sapphire, which has 
a high refractive index like ruby, is often used (A1 2 3 with Cr impurity 
is ruby, without Cr it is sapphire). In the arrangement shown in Fig. 
II.2 pump light impinging on the wide end of the sapphire trumpet is 



SAPPHIRE 




Fig. II.2 

all fed into the active ruby. As well as providing a way of getting pump 
light into the ruby, the sapphire in a composite arrangement provides a 
way for heat to escape. 

If a pulse of optical pump power is applied to a laser, then the 
number of elements in the high energy state starts to increase at the 

* "Resonant Modes in a Maser Interferometer", Fox and Li, B.S.T.J., Vol. 40, 



LASER SYSTEMS AND APPLICATIONS 



101 



beginning of the pulse and builds up until population inversion is just 
achieved, laser action occurs and the upper energy level becomes de- 
populated. If the pumping pulse is still being applied when the laser 
action is in progress then the population of the upper level might con- 
ceivably be reduced below the threshold required for oscillation and 
raised to it again once, or even several times, in the duration of the 
pumping pulse. The laser output pulse would then be expected to be 
spiky in character and this is certainly the case with many pulsed ruby 
lasers. A better-shaped pulse, and in particular a higher peak pulse 
power, can be obtained if a population inversion much greater than the 
threshold required for oscillation is established by pumping before laser 
action is allowed to start. 5 One of the mirrors could be removed or 
obscured by a shutter until a sufficiently high degree of inversion were 
achieved, or some attenuating material could be introduced into the 
optical path to reduce or "spoil" the regenerative properties, or "Q", 
of the optical cavity until the laser was required to fire. 6 A wide variety 
of different devices such as rotating mirrors, electrically operated Kerr 
cells, and mirror coatings with controlled reflection coefficients have 
been used to bring the cavity to the regenerative condition at the right 
moment. 

Practical Laser Systems 

To show the sort of apparatus used and the way it is laid out in practical 
lasers, examples of the gas, the solid, and the injection laser are briefly 
described below. In each case an attempt has been made to select a 
system which is typical of its class, but which incorporates at least 
something to minimise the defects to which its class is subject, e.g. we 
shall consider an array which reduces some of the disadvantages of the 
small emitting area of the injection laser. 

Figure II.3 shows the sort of apparatus that would be used in a typical 
CW gas laser. A quartz or silica discharge tube, about a centimetre or 
two in diameter and something less than a metre long, contains a mix- 
ture of helium and neon in proportions of about five to one with a total 
pressure of the order of 0-5 mm Hg. The helium-neon mixture will give 
laser action at about a dozen frequencies in the visible and infra-red 
regions. A given composition and pressure seems to be best for any 
particular frequency, which will be largely determined by the selective 
nature of the reflecting coating on the mirrors. The discharge tube is 

s "Q-switched Optical Masers", Midwinter and Forrester, I.E.E. Laser Sym- 
posium, September 1964. 

6 "Giant Optical Pulsations from Ruby", McClung and Hellwarth, Applied 
Optics, Supplement I, Optical Masers, 1962. 



102 



LOW NOISE ELECTRONICS 



well baked, and precautions are taken against impurities when filling it. 
The ends of the tube are closed by optical flats set at the Brewster angle 
and the light emerging and that taking part in the laser action will con- 
sequently be plane polarised. The RF discharge is maintained in the 
gas by two external electrodes wrapped round the tube and connected 
to an oscillator giving about 20 watts of power at about 30 Mc/s. The 
energy in the discharge must be sufficiently great to allow good pump- 
ing, but too much energy reduces the life of the tube which is typically 
several hundred hours. 



•1 METRE 



SPHERICAL 
MIRROR 



"1 L 



He-Ne 



'E LECTROD ES' 
=J30Mcfc|:=: 



u f 



SPHERICAL 
MIRROR 



Fig. 11.3 

The external spherical confocal mirrors have a multilayer dielectric 
coating of the appropriate material and thickness to give high reflection 
at the desired wavelength. These reflecting surfaces need to be pro- 
tected as far as possible from dirt and damage, and they will have a 
useful life comparable with that of the discharge tube. One of the 
mirrors will be arranged to transmit a small fraction of the light and this 
is the laser output which may be about 100 mW. 

The light emitting area in an injection laser is an extremely small 
strip on the face of the junction diode. If a much larger area is required, 
in order to increase the total power in the beam or to reduce its diffrac- 
tion beamwidth, then the necessary increase in junction size— par- 
ticularly the junction length— would give rise to great difficulties because 
of the large driving current needed and because of the complications in 
the collimating optics. 

An injection laser array has been produced 7 where the drive current 
is passed in series through a number of diodes which each has its own 
collimating lens. In Fig. II.4 the diagram is a section, the light emitting 
junction being virtually a line and the lens cylindrical. With ten laser 
diodes, mounted on copper blocks and connected thermally to a heat 
sink cooled with liquid nitrogen, 400-watt peak light output pulses were 

1964* Semiconduct0r Laser Arra y"' Broom, LEE. Laser Symposium, September 






LASER SYSTEMS AND APPLICATIONS 



103 



obtained with 200-ampere peak driving current pulses of length 0*7 
microsecond, 




WAWAW 



JUNCTIONS 



T77??>f?t? 



LIGHT ■* y- 

LENSES 
\ 

LiGHt >_ ~D 



COLLIMATED 

LIGHT 

OUTPUT 



Fig. II.4 

In the case of the ruby or other solid laser 8 there are really two classes 
of practical system to consider: there is the system designed to give CW 
operation 9 and the "giant pulse" system which aims at the largest 
possible peak pulse output. In each case there is a general requirement 
for a high quality crystal laser rod and good mirrors. But in the CW 
case there is concentration on continuously operating high energy 
pump lamps and techniques, like sapphire sheathing for the active rod 
and liquid nitrogen cooling, to get the most efficient use of pump 
energy 10 and the best removal of heat. In the giant pulse systems 11 the 
laser action is held off until the highest state of population inversion is 
achieved and much interest is concentrated on the various types of 
"Q spoiling" like the spinning prism, the Kerr cell, or the synchroinsed 
ultrasonic shutter. 12 

Applications of Laser Action 

The applications of the laser tend to occur at three levels of increasing 
sophistication. There is, first of all, the ability to concentrate very high 
power on to a very small spot; this may be used in machining or certain 
research where very high local temperatures are required. Second in 
sophistication is the use of the fact that the high power is at a single 
well-defined frequency; this may be used in more selective atomic or 
molecular excitation processes, particularly in fundamental research but 

8 "Crystalline Solid Lasers", Kiss and Pressley, P.I.E.E.E., Vol. 54, October 1966. 

9 "Excitation, Relaxation, and Continuous Maser Action in Calcium Fluoride", 
Boyd et al., Phys. Rev., Letters, Vol. 8, 1962. 

10 "Pumping Power Considerations in an Optical Maser", Svelto, Applied Optics, 
Supplement 1, Optical Masers, 1962. 

11 "Giant Optical Pulsations from Ruby", McClung and Hellwarth, J.A.P., Vol. 
33, 1962. 

12 "Ultrasonic Refraction Shutter for Optical Maser Oscillator", De Maria et al., 
J.A.P., Vol. 34, March 1963. 



104 



LOW NOISE ELECTRONICS 



also in the optical pumping of masers. Most advanced of all is the use 
of the fact that the high power is at a very high frequency and hence can 
provide an enormous bandwidth when modulated in a communication 
system. 

These applications are briefly discussed below with references to 
papers giving more detailed descriptions. Developments are, however, 
proceeding so rapidly that the current journals will almost always 
contain papers of importance which make out of date the performance 
figures published in textbooks. 

Laser as Cutting or Welding Tool 

One of the most frequently exhibited laser demonstrations is the 
punching of a hole in a razor blade by a focused laser pulse which may 
have a power density of 10 4 or 10 5 Mw/cm 2 . in a spot of diameter 
50 microns or less. The possibilities of such a precise tool in micro- 
machining are fairly obvious, but in such an application the regular 
pulses at fairly high repetition rate available from a pulsed gas laser are 
more likely to be used than single giant pulses from a spoiled Q solid 
laser. 

For machining, the material must be removed by melting and sub- 
sequent vaporisation, but these processes when brought about by a laser 
pulse are of considerable complexity and much must be discovered 
about them before machining is widely carried out with the laser. 13 
At a lower power level melting alone may be achieved and this has been 
used in welding processes in such applications as microminiature 
electronic circuits. The laser tool has perhaps been most developed in 
the medical field, particularly in eye surgery 14 and in fairly conventional 
radiotherapy of the X-ray type. 

Fundamental Investigations 

The laser is a very high-powered source of radiation at a single optical 
frequency, or at any rate spread over a very narrow frequency band. 
When focused, the power available per unit area of target is higher by 
many orders of magnitude than ever before known and most applica- 
tions of the laser in fundamental science exploit this previously un- 
obtainable level of power. In the most direct way, the power can be 
used to produce very high temperatures where normal heating is 
inconvenient and work of this sort has been carried out on high 

13 "Vaporisation by Laser Beams", Hughes, I.E.E. Laser Symposium, September 
1964. 

14 "Use of Ruby Laser for Retinal Photo-coagulation". Smart, I.E.E. Laser 
Symposium, September 1964. 



LASER SYSTEMS AND APPLICATIONS 



105 



temperature plasma. 15 Such very high energy environments, with 
electromagnetic fields at well defined frequencies, may also be 
important in the investigation of chemical reactions. 

There is a number of optical processes in which the frequency of the 
light involved is changed. Research in this field is very important 
because it is expected to yield information about fundamental radiation 
processes and about the structure of matter. In these processes the 
amount of light suffering such a change of frequency is very small and 
the experimental work has been greatly restricted by the difficulty of 
detecting it. Indeed, the theory of some of the most interesting pheno- 
mena which had been predicted, showed that there was no possibility 
with existing instruments of detecting the small amounts of light in- 
volved. With the large laser power outputs available, detection is 
simplified in those cases like Raman scattering where the effect had 
already been observed, and made possible for the first time in the case 
of the non-linear processes. 

Raman effect studies once involved long photographic exposure 
times to detect the scattered light, but the experiments are now greatly 
aided and extended by the monochromatic light of high power available 
from the laser 16 . The Raman scattered light from organic liquids, which 
is monochromatic but different in frequency from the original laser 
light, may in itself constitute a useful radiation source and more 
recently it has been shown 17 that stimulated emission can occur in 
suitable Raman sources. 

The theoretical prediction of multiquantum effects 18 in certain 
crystals has been verified using laser outputs. 19 Typically two photons 
may be absorbed simultaneously to raise the material from a low to a 
higher energy level. A subsequent downward transition between the 
same levels may occur with the emission of a single photon at twice the 
frequency of the two absorbed photons. Practical applications in 
harmonic generation and mixing are likely to be very important. 20 



Communication and Ranging Systems 

The frequency of the radiation from a laser is of the order of a million 
times that used in communication systems at present and, in theory at 
any rate, a laser communication link could carry a million times more 

» Minck, J. App. Phys., Vol. 35, 1964. 

19 "Ruby Optical Maser as a Raman Source", Porto and Wood, /. Opt. Soc. Am., 
Vol. 55, 1962. 

17 Woodbury, Proc. I.R.E., Vol. 50, 1962. 

18 Loudon, Proc. Phys. Soc, Vol. 80, 1962. 

19 Giordmaine, Phys. Rev., Letters, 8, 1962. 

20 Bass et at., Phys. Rev., Letters, 8, 1962. 



106 



LOW NOISE ELECTRONICS 



channels than a conventional one. Because the wavelength is so small 
the divergency of the laser beam will be very small indeed, and a re- 
flector of the same sort of size as a microwave aerial would, again in 
theory, give extremely high directivity and corresponding sensitivity, 
which would imply long detection ranges in optical radar and com- 
munication systems even with quite low-powered sources. 

It is as well to reconsider the enormous theoretical advantages of 
optical communication systems in more practical terms at an early 
stage. Although such considerations may modify early enthusiasm for 
laser systems, it is worth remembering that the failure to achieve the 
theoretical performance figures is because of inadequacies in "hard- 
ware" which is in the very earliest stages of development and may show 
great improvements soon. 

Although the short wavelength of the optical radiation means an 
extremely narrow beam from a radar size reflector, there is an implica- 
tion that such a reflector can be made to optical tolerances and that it 
will retain its shape and surface qualities in use. Furthermore, although 
a very narrow beam will give high sensitivity, it will be extremely 
difficult to hold on the target and some auxiliary tracking system may 
well be needed. 

The full realisation of the very large bandwidth theoretically possible 
with a laser system depends upon certain practical considerations. A 
suitable efficient method of modulating the light beam must be available 
and such modulators are only just being developed. They are usually 
electro-optical devices 21 like Kerr cells operating on the tight beam for 
ordinary lasers, and simpler electrical modulators operating directly on 
the driving current in the injection laser. Even if the modulation is 
efficient, the frequency stability of present lasers, particularly those of 
high power, will have to be much improved if many closely spaced 
communication channels are to be handled on the same carrier. In 
addition, a superhet receiver must be developed if the possibilities of 
the coherent system are to be fully exploited. 22 

Assuming that suitable terminal equipment is designed for an 
operational laser communication system there are still problems 
associated with propagation through the intervening medium. At 
optical and near infra-red frequencies the attenuation in a clear at- 
mosphere is fairly great, and under foul weather conditions it will rise 
considerably. The choice of operating frequency must be made with 

21 "Electro-optical Modulation at Microwave Frequencies", Harvey, I.E.E. Laser 
Symposium, September 1964. 

22 "Superheterodyne Reception at Optical Frequencies", Warner and Warden, 
l.E.E. Laser Symposium, September 1964. 



LASER SYSTEMS AND APPLICATIONS 



107 



care to avoid peaks in the absorption spectrum, and even if the signal 
intensity is adequate, there may well be phase and path variations due 
to turbulence which will degrade the system performance. 23 Since 
commercial laser communication systems would presumably be of very 
large capacity it may well be economic to use guided propagation in 
glass fibre pipes or cables. 24 - 25 

Pilot models of laser communication equipment have been built 
using ordinary and injection lasers, while useful range-finding equip- 
ment for short-range military tasks and for long-range space applica- 
tions also exist. Progress is rapid and the current journals or the report 
of the latest laser conference will provide the best information on new 
applications. 

23 "Unguided Optical Propagation in the Atmosphere and Undersea", Meredith, 
I.E.E. Laser Symposium, September 1964. 

24 "Communication Systems in the Visible and Infra-red Spectra: Present and 
Future", Karbowiak, I.E.E. Laser Symposium, September 1964. 

25 "Optical Transmission Research", Miller and Tillotson, P.I.E.E.E., Vol. 54, 
October 1966. 



I 



Appendix III 

Electron Beam Amplifiers 



in many electronic devices, including the conventional valve, ampli- 
fication is achieved by converting some of the D,C. energy of an electron 
stream into A.C. energy in a suitable circuit. In the following section 
we are concerned with long well-focused beams used in the Gc/s region 
in such devices as the klystron and the travelling wave tube. 

Attempts to understand electron beam tubes in terms of the ballistics 
of single electrons are much complicated by the effects of space charge. 
If the electron beam is considered as an elastic medium which resists 
displacement of the electrons from their uniform density, then useful 
information about these devices can be obtained in terms of the 
wave-like behaviour of compressions and rarefactions in the electron 
beam. 

Space Charge Waves 

Let us first reduce the problem to a single dimension by considering the 
simplest possible beam consisting of electrons moving in single file 
through a perfect vacuum. Originally the electrons all move at the same 
velocity, but modulation may be imposed on the beam by passing it for 
a short part of its path through a small A.C. electric field. Successive 
groups of electrons passing through the field will be retarded, unaffected, 
and accelerated as the electrostatic force due to the alternating field 
changes from a maximum in the direction opposite to the electron 
velocity, through zero, to a maximum in the direction of the beam 
velocity. 

After leaving the field the electrons will have different velocities. 
Those retarded will have low velocities and will fall back towards the 
unaffected electrons immediately behind them, while these unaffected 
electrons will also be approached from the other side by the accelerated 
electrons now moving with high velocities. Some distance away from 
the alternating field the electron beam, originally of uniform density, 
will consist of high density bunches separated by low density gaps, the 



ELECTRON BEAM AMPLIFIERS 



109 



centre of each bunch and gap being an unaffected electron which passed 
through the A.C. field when its value was zero. 

If this simple case is considered in detail it can be shown, e.g. by 
using an Applegate diagram, 1 that optimum bunching occurs at some 
point a certain distance from the alternating field. Many of the electrons 
which passed in succession through this field will arrive at the optimum 
bunching point simultaneously. 

The effect of the velocity modulation imposed by the alternating field 
is to produce bunches and gaps, i.e. the electron beam, originally of 
uniform density, is compressed and rarefied. The electrostatic force of 
repulsion between electrons will, however, tend to oppose such com- 
pression and rarefaction and will act like the restoring force which 
resists displacement in an elastic medium. 

Any disturbing force tending to modulate the uniform beam is 
opposed by the electrostatic force tending to reduce to zero the displace- 
ment of electrons from their equilibrium positions. The result of any 
disturbance will be an oscillation in which the electron displacement 
varies periodically in a manner determined by the general properties of 
the electron beam. 

It is fairly obvious that there is a possibility of wavelike behaviour if 
the electron beam is disturbed. We shall not investigate this idealised 
case any further but shall consider similar oscillations in the more 
complex beams of electrons and positive ions which are used in practical 
devices. These oscillations are called plasma oscillations. 



Plasma 

A plasma is a region where there is a large concentration of ions equally, 
or nearly equally, divided between positive and negative so that the 
space charge is approximately neutral. Un-ionised molecules will also 
usually be present in a plasma, which may occur in a gas, a liquid, or a 
semiconducting solid. 

We shall be concerned with the gaseous plasma which has been very 
extensively investigated because of its occurrence in many fields of great 
theoretical and practical importance, such as the gas discharge, the 
ionosphere, the thermonuclear fusion reactor, and the microwave beam 
amplifiers which are our particular concern. 

There is sufficient residual gas in a "hard" vacuum tube for a beam 
of fast-moving electrons to produce considerable numbers of positive 
ions. The positive gas ions tend to move along the axis in the low 
potential region due to the electrons, although some escape from the 

1 Electron Physics and Technology, Thomson and Callick, E.U.P., 1959. 



110 



LOW NOISE ELECTRONICS 



beam and reach the walls of the tube where they are neutralised. In 
general, the rate of production of positive ions is sufficiently high for 
the space charge in the beam to be neutralised enough to enable it to be 
treated as a plasma. 

For an idealised beam which is completely neutralised, of infinite 
cross-section, and with small axial variations in charge density due to 
oscillation, it is possible to calculate the natural radian frequency (o) p ) 
of plasma oscillation. 2 



h Po 



where e and m are the electronic charge and mass, po is the electron 
charge density in the beam, and eo is the permittivity of free space. 

In the real electron beams which occur in practical tubes the ideal 
plasma conditions assumed above do not hold. The beam is of finite 
cross-section, and radial vibrations as well as axial ones are possible. 
A number of modes and frequencies are possible in a real beam, but 
usually only the fundamental is of any significant amplitude. 

The frequency of the fundamental depends to a certain extent on the 
geometry and the operating conditions of the tube, but it is close to — 
only less than — the frequency calculated for the ideal case above. 

In future we shall assume that there is no significant difference be- 
tween these two frequencies and we shall use w p to refer to the funda- 
mental plasma frequency of a real beam, sometimes called the reduced 
plasma frequency. 

Disturbance Imposed on Beam 

Suppose we have an electron beam with a natural plasma frequency a> p 
and that we impose upon it a disturbance due to a signal at a frequency 
«. This might be achieved by allowing the beam to pass through an 
alternating electric field between two closely spaced wire meshes to 
which a small r.f. voltage at frequency o> is applied. This alternating 
field will accelerate or retard each electron as it passes between the 
meshes of the buncher and velocity modulation will be produced. It is 
convenient to consider the resultant complex electron behaviour in 
terms of space charge waves produced on the beam. 

Each electron passing through the buncher receives some sort of 
impulse from the applied field (&>) and will suffer a displacement from 
its equilibrium position in the beam. Because it has been so disturbed 
it will oscillate about its equilibrium position at the natural plasma 

2 Physical Electronics, Hemenway et al., Wiley, 1962. 



ELECTRON BEAM AMPLIFIERS 



111 



frequency m p , while the equilibrium position travels at the beam 
velocity w . Although each electron is oscillating at frequency w p , the 
amplitude and phase of the electron displacements depend on the size 
and direction of the impulse received from the modulating field, i.e. it 
depends upon w. 

If this complex motion is analysed it is found 3 that there are two 
space charge waves on the electron beam which are important in the 
operation of amplifying valves, and all other modes are neglected. 

These two waves are called the fast and the slow waves and they have 
phase velocities (ty and v$) which are typically a few per cent higher and 
a few per cent lower than the beam velocity. 



Vf m 



Uq 



1 _ ^1 



V, = 



Uq 



1+^ 
«a 



The phase constants of the two waves are given by p = - — and 

the group velocity ( , J is in each case uq the beam velocity. Thus 

the fast wave has a phase velocity greater than the group velocity, while 
the slow wave has a phase velocity less than the group velocity. 

Fast and Slow Wave 

The reader will be familiar with examples where complex periodic 
properties are analysed into a number of sinusoidal components which 
are considered separately. This approach is used when the example of 
the last section is considered in more detail. 

When a disturbance at frequency a is applied to the electron beam 
then a complex periodic electron axial motion is produced which can 
be analysed into two components: the fast wave and the slow wave, each 
of frequency « but of different phase velocities. 

Thus the fast wave will consist of compressions and rarefactions of 

the beam which travel down the tube at velocity v* = - — The 

1 — — 
ta 

separation between successive compressions will be v/l =- • 

Because these compressions and rarefactions move at a greater 
velocity than the beam velocity the establishment of a fast wave means 
that extra energy has been fed into the beam by the source producing 
the wave. 

3 "Wave Picture of Microwave Tubes", Pierce, B.S.T.J., Vol. 33, November 1954. 



112 



LOW NOISE ELECTRONICS 



The slow wave also consists of compressions and rarefactions, but 
these travel at v s so that the separation between successive compressions 
2^v s /bi is less than that in the fast wave. In this case the compressions 
and rarefactions travel at less than the beam velocity and, to produce a 
slow wave, energy must be extracted from the beam by the slow wave 
source. 

It is worth noting that radio-frequency energy at frequency m must 
be fed into a beam if the amplitude of a fast wave at that frequency is to 
be increased, whereas energy must be extracted if the amplitude of a 
slow wave is to be increased. 

The unusual general properties of the fast and slow wave are well 
illustrated in the klystron amplifier. 

Klystron Amplifier 

In the simple klystron amplifier shown in Fig. III. 1 an electron 
beam moves from left to right passing through wire meshes in the two 
resonant cavities at A and B. The cavities are tuned to the signal fre- 
quency ti> and the alternating electric field between the two meshes in 
each cavity will impose a disturbance on the beam and produce waves 
of the type described earlier. 



SIGNAL U) IN 



AMPLIFIED 
SIGNAL OUT 



SEAM. 



(LJ 



Fig. III. I 

The beam to the left of the first cavity (buncher) is homogeneous and 
when it passes through the electric field in this first cavity each electron 
will receive some sort of impulse from the field and will oscillate axially 
at frequency wj,. We have already seen that the phase and amplitude 
relation between the displacements of individual electrons is complex. 
The complex periodic motion may be analysed into two waves of equal 
amplitude : the fast wave and the slow wave, each consisting of com- 
pressions and rarefactions of the electron beam. Since the two waves 
are of equal amplitude the buncher feeds no nett power into the beam. 



ELECTRON BEAM AMPLIFIERS 



113 



The compressions travel down the tube faster than the beam velocity w 
in one case and slower than wo in the other case. 

Immediately after leaving the first cavity the electrons will all have 
different velocities, but there will have been no time for any overtaking 
to occur, so the beam is still of uniform density with no bunching. Thus 
the compressions and rarefactions due to the fast wave exactly cancel 
those due to the slow wave at A, i.e. the two waves are exactly out of 
phase. 

Because the two waves have different phase velocities, v f and v s , they 
will change phase relative to each other as they travel down the tube 
until at B, a distance rffrom A, a relative phase change of * has occurred 
and the two waves will be in phase, with compressions due to the slow 
wave adding to those of the fast wave to produce maximum bunching. 

Phase constant p = — is phase change per unit distance travelled 

A 

__ 2tc x frequency _ <>> 

velocity _ velocity 



But 



velocity — 



o>Uq 



ui Up 



W — 6>„ . (i)-j-(Dp 

Pfaat = — - — - and pgiow = — 

Uq Uq 

Relative phase difference introduced per unit distance travelled is 

2c>p 
PbIqw — Pfast = — - 
Wo 

For a phase difference of % to be introduced the distance travelled is 



d = 



2o) P /wo 



2w„ 



d is the optimum bunching distance and the second cavity (catcher) is 
placed a distance d from the buncher. 

Radio frequency oscillations will be induced in the second resonant 
cavity (catcher) by the bunched electron beam passing through it. The 
alternating electric field between the wire meshes of the cavity will 
impose a disturbance on the beam and generate a fast and a slow wave at 
frequency u just as the buncher did. 

For amplification to occur the beam must lose energy and the catcher 
cavity must gain it. Such a transfer of energy will occur due to inter- 
action of the beam with the alternating electric field if the cavity is 
tuned so that the radio-frequency field has the correct phase. 



114 



LOW NOISE ELECTRONICS 



Fast wave amplitude increases when energy is fed into the beam and 
slow wave amplitude increases when energy is extracted. Thus to ensure 
a transfer of energy from beam to catcher, the fast wave generated by 
the second cavity must be of such a phase that it subtracts from the 
original fast wave amplitude due to the buncher, while the new slow 
wave will add to the original one. 

The beam to the right of B is still inhomogeneous but with a reduced 
fast wave amplitude, and an increased slow wave amplitude, the energy 
extracted from the beam to produce these changes appearing as radio- 
frequency energy in the cavity. 

Travelling Wave Tube 

In the klystron amplifier the two circuits, buncher and catcher, interact 
with the electron beam at two single points on the axis. In the travelling 
wave tube, and in the backward wave tube which is described later, 
there is a single, long, distributed radio-frequency circuit which interacts 
with the electron beam along its whole length. The electromagnetic 
wave travelling along the distributed circuit reacts with one of the waves 
on the beam. For this to happen the circuit wave and the selected beam 
wave must have approximately equal velocities. 
The two beam waves will have velocities 



v = 



«o 



1 ± Wj,/fc> 



which are of the same order as the electron beam velocity wo. This must 
be very much less than the velocity of light because of the difficulty of 
accelerating electrons to anything approaching that speed with normal 
H.T. voltages. 
With a beam voltage V of 3000 volts the electron velocity «o is given by 

Ve = ±mu Q 2 
ejm = 1-76 x 10 u coulomb/kilo. 



= l 2Ve m 
V m 



wo = J — = (2 X 3000 x 1-76 X 10")* = Vl0*5 X 10 7 m/sec 

Thus the beam velocity is little more than a tenth of the velocity of 
light and the distributed circuit will have to be one in which the velocity 
of electromagnetic waves is reduced to about this value. 

If a waveguide is corrugated or loaded internally with discs or posts, 
then the wave velocity is reduced. The simplest slow wave structure is, 
however, the wire helix, where the wave travels down through the helix 



ELECTRON BEAM AMPLIFIERS 



115 



at roughly the same speed as if it were actually travelling along the full 
length of the wire. 
Thus if the helix has a diameter of 0-1 in. and there are 30 turns to the 

c 
inch, then the wave velocity down the helix is about , , ^ 

i.e. little more than a tenth of the velocity of light. 

In the detailed discussion of travelling wave tube amplification we 
shall see that the circuit wave is required to interact with the slow 
electron beam wave and not with the fast one. Thus the circuit wave 
and the slow electron beam wave must have about equal velocities. 



ANODES 
CATHODE | 



P 



FOCUSSING MAGNETIC FIELD 
CIRCUlfwAVE 



BEAM 



COLLECTOR 



R.E SIGNAL IN 



AMPLIFIED 
SIGNAL OUT 



Fig. III.2 



The electron beam is accordingly designed so that <* v is large enough 
to give good separation between the velocities of the slow and fast 

wo 



electron beam waves 



v = 



i 



The helix is then designed to 






1 ±o>p/ci)_ 

make the circuit wave velocity synchronise with the slow beam wave and 
not with the fast one. 

In Fig. III. 2 the beam produced by the electron gun is shown (full 
line arrows) entering and leaving the helix. To avoid confusing the 
diagram the arrows are not continued through the helix. The dotted 
arrows show the radio-frequency field of the circuit wave passing down 
the helix from left to right. 

A steady axial magnetic field focuses the electron beam so that it has 
a radius comparable with that of the helix, but is not intercepted by it at 
any point. 

When the radio-frequency signal is coupled into the left-hand end of 
the helix it reacts with the electron beam to produce a slow beam wave 
provided that the helix has been designed, and the tube current and 
voltage adjusted, to give synchronism between the circuit wave and the 
slow beam wave as described above. 

The amplitude of the slow wave on the electron beam will gradually 
increase as it moves from left to right. But we have already seen that to 
increase the amplitude of the slow wave power must be extracted from 
the beam. In going from left to right the power in the slow wave 



116 



LOW NOISE ELECTRONICS 



decreases — its negative power increases — and the positive power in the 
coupled circuit wave increases. 

In fact, the total power contained in the coupled wave system re- 
mains constant 4 with the circuit wave growing exponentially from its 
original value and the beam wave "growing" similarly in the negative 
sense (Fig. III. 3). 



CIRCUIT WAVE 



DISTANCE ALONG HELIX 




BEAM WAVE 



Fig, III.3 

Amplification occurs along the whole length of the helix and the gain 
is high even although the circuit is non-resonant. The phase velocity 
of the wave on the helix is almost constant over a wide frequency range 
and the travelling wave tube has a very large bandwidth. 

The great bandwidth, which might be say 1 Gc/s at 3 Gc/s, means 
that the tube is very useful in microwave links carrying many television 
or telephone channels. The original travelling wave tubes in general 
use fifteen years ago had noise temperatures of a few thousand degrees, 
but more recent techniques discussed below for limiting noise on an 
electron beam have given tubes with noise temperatures of a few 
hundred degrees. 

The high gain over a large bandwidth causes problems of instability 
and oscillation because the coupling at the output end will certainly not 
be a perfect match at all frequencies. There is a possibility that the 
energy reflected back down the helix to the input may cause oscillation. 

To avoid this oscillation, attenuation is introduced about halfway 
along the tube by spraying graphite over a short length of the glass 
which supports the helix. The currents induced in the graphite by the 
electromagnetic wave on the helix cause the loss. 

Suppose the tube has a forward gain G in the absence of attenuation. 

Let the attenuation be A. 

An input signal of nominal value 1 appears at the output as G(A 
which is the effective amplification of the tube. 

4 "Low Noise Amplifiers for Centimetre and Shorter Wavelengths", Wade, 
P.I.R.E., May 1961. 






ELECTRON BEAM AMPLIFIERS 



117 



Suppose a fraction k of the output is reflected by a mismatch, then 
the signal leaving the output towards the input end is kGjA. 

This wiU suffer attenuation and the signal arriving at the input is 
kG/AK 

No oscillation will occur if kG/A 2 < 1. 

Thus if a tube has 70 db of unattenuated forward gain, then a 40 db 
attenuator will certainly prevent oscillation while leaving an effective 
gain of 30 db. 

Backward Wave Tube 

Certain filter type circuits will allow the passage of waves which have 
their phase and group velocities in opposite directions. Such a wave is 
called a backward wave, 5 and if it is coupled to the slow electron beam 
wave then power transfer from the beam to the circuit takes place. 



OUTPUT 



INPUT 
V 



CIRCUIT 



BEAM 



ELECTRON 
GUN 



COLLECTOR 



Fig. III.4 






Suppose that a signal is introduced into the backward wave circuit 
at A (Fig. III.4), then the energy will travel down the circuit from right 
to left until it reaches the output end where it is removed to an external 
load. The phase velocity of this circuit wave is, however, from left to 
right, and if the power supplies are adjusted so that the slow beam 
wave couples with the circuit wave, then the amount of signal energy 
reaching the output end is greater than that entering at A. A special 
feature of the backward wave tube is the existence of positive feedback 
because the circuit wave at the output induces the slow wave on the 
electron beam which travels down towards the collector end of the tube 
where it induces a wave on the circuit at A in phase with the input 
signal. 

Circuit waves of different frequency have different phase velocities 
and only one such wave will be in synchronism with the slow beam 
wave for a given beam velocity. Changing the beam voltage and hence 
the beam velocity will alter the tuning of the tube. 

As described the backward wave tube could be used as an amplifier 
of narrow bandwidth with the input applied at A and the output taken 

5 Travelling Wave Tubes, Pierce, Van Nostrand, 1950. 



118 



LOW NOISE ELECTRONICS 



from the circuit at the electron gun end. 6 If, however, the beam current 
exceeds a certain value called the starting current the tube will oscillate 
and it is as a backward wave oscillator that it is generally used. 7 - 8 There 
will be no input at A, and the output will be taken from the gun end as 
shown. 

The great advantage of the backward wave oscillator is that it can be 
tuned over a wide range, say 7-12 Gc/s or 2-4-4-5 Gc/s, by varying the 
beam voltage. If the beam current is greatly increased, then oscillations 
at higher frequencies than the fundamental are generated simul- 
taneously. 

Noise Redaction 

Most of the noise in an electron beam tube arises from shot noise pro- 
duced by the cathode. This is greatly reduced if a well-designed electron 
gun is used. In particular, modern low noise tubes have multi-anode 
electron guns, with the shape and relative potentials of the anodes 
chosen as a result of a complex calculation which often justifies the use 
of analogue computation methods. 9 Commercial tubes will have a 
variable voltage supplied to at least one of the anodes and this is 
adjusted in operation to give minimum noise. 

Further reduction in noise can be achieved by a sophisticated 
technique of stripping the residual noise power from the beam before it 
enters the interaction region. 

We have already seen that coupling between the circuit wave and the 
slow beam wave results in an exponential growth of the amplitude of 
both waves (p. 116). The positive power in the circuit wave increases 
and the negative power in the slow beam wave increases, the total power 
in the system remaining constant. 

If the circuit wave is coupled to the fast beam wave, then the total 
power remains constant as before but there is a gradual interchange of 
power between the circuit and the beam. If a particular short length of 
circuit is chosen, then, by the time the beam reaches the end of the 
circuit, all the power entering on the circuit will have been transferred 
to the beam, and all the fast wave power entering on the beam will have 
been transferred to the circuit and may be led away. 

6 "Crossed-field Backward Wave Amplifier. Noise Figure Studies", Mantena and 
Van Duzer, J. Electronics and Control, Vol. XVII, No. 5, November 1964. 

7 "Analysis of the Backward Wave Travelling Wave Tube", Heffner, P.I.R.E., 
June 1954. 

8 "Backward Wave Oscillators", Johnson, PJ.R.E., June 1955. 

9 "Analogue Method of Determining Electrode Shapes of Electron Guns", 
Lomax, J. Electronics and Control, Vol. XV, No. 3, September 1963. 



ELECTRON BEAM AMPLIFIERS 



119 



The electron beam may be passed near such a short length of circuit 
immediately after leaving the gun and before it enters the main inter- 
action region. The beam will be carrying both slow and fast wave noise 
power on leaving the gun. The signal is applied to the start of the short 
length of circuit. At the end of the circuit all the signal power will be 
on the fast beam wave and alt the fast noise power will have been re- 
moved from the beam into the circuit and led away. 

We now have a noise-free fast wave carrying signal power ready to be 
amplified, but we also have a slow wave carrying noise. If the beam 
velocity is now suddenly increased the slow wave will be eliminated 10 
leaving a noise-free fast wave carrying signal power to be amplified. 

The amplification process cannot, however, be the ordinary travelling 
wave tube technique described earlier because this uses a slow beam 
wave and we have launched a noise-free fast wave. Amplification using 
a fast wave can be achieved by supplying power at twice the signal fre- 
quency. Beam tubes using this principle are called electron beam para- 
metric amplifiers and are discussed elsewhere (p. 81). 

Practical Electron Beam Tubes 

Descriptions of the structure of practical electron beam tubes are 
available in such references as 11 ' 12 - 13 , and manufacturers' data sheets 
give performance figures. Such literature will contain details of beam 
voltage and current, power output, gain, noise figure, and tuning range, 
focusing magnetic field and whether it is provided by a solenoid or 
permanent magnet system. 14 Waves, usually of fairly low frequency, 
associated with the heavy gas ions in a plasma can have analogous 
behaviour to the electron beam waves already described, and some of 
their properties are outlined in reference 10 . 

In general low noise, wide-band, low power travelling wave tubes 
are now available with noise temperatures of about 500 degrees ab- 
solute, and tubes with higher noise temperatures may give many watts 
output with large bandwidth. 15 For very high power output multi- 
cavity klystrons several feet tall have high efficiency and give many 
kilowatts of mean power. 

10 "Role of Space Charge Waves in Modern Microwave Devices", Sims and 
Stephenson, Electronic Engineering, July 1960. 

11 Microwave Engineering, Harvey, Academic Press, 1963. 

12 'Travelling Wave Tubes for Communications Networks", Coulson, Marconi 
Point to Point Telecomms., Vol. 4, No. 2, February 1960. 

13 "Backward Wave Oscillator for Millimetre Waves", King, Marconi Renew, 
Vol. XXV, No. 146, 1962. 

14 "Compensated Reversed Field Focussing of Electron Beams", Burke, PJ.E.E.E., 
Vol. 5 J, No. 11, November 1963. 

18 Power Travelling Wave Tubes, Gittins, E.U.P. (1965). 
E 



120 



LOW NOISE ELECTRONICS 



ELECTRON BEAM AMPLIFIERS 



121 



Crossed Field Tubes 16 

In the ordinary (or 0-type) electron beam tubes discussed above the 
power required for the amplification process is drawn from the kinetic 
energy of the electrons. These electrons must be kept in approximate 
synchronism with the R.F. wave travelling on the helix or other wave- 
supporting structure. Consequently there will be considerable energy 
left in the beam when it has ceased to interact with the wave, and O-type 
tubes will be of low efficiency. 

If electrons are injected into a region where there are crossed electric 
and magnetic fields, then the electron paths become complex, but the 
drift velocity of the electrons through the region, which also contains 
the wave-supporting structure, remains fixed and is independent of the 
electron velocity. 

By a suitable choice of the crossed electric and magnetic fields the 
electron drift velocity can be made to synchronise with the structure- 
borne R.F. wave. Even although this wave receives much energy from 
the electrons (which in turn receive it from the electric field), syn- 
chronism will still be maintained, regardless of the length of the inter- 
action region. 

Amplifiers and oscillators using crossed fields in this way are called 
Af-type tubes. They may be forward or backward wave operating and 
are very much more efficient than O-type tubes and are capable of very 
high power operation. M-type tubes have, however, poor noise 
qualities and, while they deserve mention in a general appendix of this 
sort on electron beam tubes, they have no low noise application. 

Although il/-type operation can be used with a long tube of the con- 
ventional travelling wave type, it is much more convenient to bend the 
wave-supporting structure into a circle and have a more compact 
magnetic field. The most familiar example of the Af-type tube is 
probably the magnetron, a forward wave oscillator, which was invented 
before Af-type operation was fully understood. 

If the magnetron anode structure, which supports the R.F. wave, is 
broken to allow an input and an output, then the valve may be used as a 
backward wave amplifier (amplitron). If an external frequency-deter- 
mining feedback circuit is connected between output and input then a 
backward wave oscillator (stabilotron) is obtained. Both these tubes 
are increasingly used in commercial and military systems. 

The amplitron allows a complex signal, such as the pulse of a 
sophisticated radar system, to be generated at low level and then 
amplified with good efficiency to a high power level. The stabilotron is 

10 Crossed Field Microwave Derives, Okress, Academic Press (1961), 



tending to replace the magnetron because of its greater frequency 
stability and efficiency. 

Another M-type backward wave oscillator is the carcinotron which 
has a localised cathode rather than the extended cylindrical cathode of 
the magnetron and stabilotron. Very broad band tunable operation at 
high power is possible with the carcinotron which is likely to be princi- 
pally used in electronic warfare as a jammer, where the noisy character 
of the output is little disadvantage. 






Appendix IV 

Ferrites and their Microwave 
Applications 



the ferrites available since 1946, are a special group of magnetic 
materials with very high resistivity. Because of their high resistance, 
eddy current effects are much reduced and ferrites can be used as cores 
for transformers and coils in the high frequency range where conven- 
tional metal cores in laminated or dust form become inefficient. 

The origin of the magnetism in strongly magnetic materials is the 
spinning electron. If an electromagnetic wave is propagated through a 
magnetic medium, then there is interaction between the spinning 
electrons and the alternating magnetic field of the propagated wave. 
This interaction is strongest at frequencies in the microwave region but 
in metals eddy currents cause the skin depth to be very small. 
The high resistance of the ferrites, however, means that the alternating 
magnetic field wilt penetrate them in depth, even at ultra-high fre- 
quencies, so that the effects of this interaction between a microwave 
signal and the electron spin can be readily observed and used. 

It can be arranged that the interaction between the microwave signal 
and the electron spin in the ferrite results in such effects as the absorp- 
tion of energy from the signal or in the introduction of a phase change. 

The character and amount of the interaction between signal and spin 
is affected if a steady magnetic field is also applied to the ferrite. 
Variation of this steady magnetic field can thus be made the means of 
producing a variable attenuator or a variable phase changer in a wave- 
guide system. 

The nature of the interaction between the electron spin and the 
microwave signal depends upon the relative directions of the polarisa- 
tion of the signal and the steady applied magnetic field. For a given 
microwave signal, propagation through a piece of ferrite with a steady 
applied magnetic field will produce one effect for propagation in one 
direction, and a different effect for propagation in the opposite direction. 



FERRITES AND THEIR MICROWAVE APPLICATIONS 



123 



Such unidirectional effects enable the microwave isolators, circulators, 
etc., which are described below, to be realised. 

In Fig. IV. 1 the isolator represents an element which gives complete 
attenuation for signals passing from left to right, but zero attenuation 
for similar signals passing from right to left. 



/vw\ 



IT 



ISOLATOR 



GYRATOR 
Fig. IV.l 




CIRCULATOR 



In the gyrator a phase change of x radians is introduced into signals 
going from left to right, but no phase change for signals going from 
right to left. 

In the four-port circulator signal can only pass from any one port to 
its next neighbour in the clockwise direction. Passage from 1-2 or 2-3 
or 3-4 or 4-1 is possible, but infinite attenuation is encountered in any 
other path such as 1-4 or 3-1. 

Structure, Resistance, and Magnetism 

The ferrite molecule has the general formula MFe204 where M is some 
divalent metal such as nickel, cobalt, or magnesium, and the two iron 
atoms are trivalent ferric atoms. In the naturally occurring ferrite 
Fe304 (magnetite or ferrous ferrite) M is a divalent ferrous atom. 

A finely powdered intimate mixture of metal carbonates or nitrates is 
prepared so that the correct proportions of M, Fe, and O atoms will 
appear in the ferrite molecule after chemical combination. The 
chemical combination and sintering may be carried out at the same 
time by heating the powder to a high temperature under pressure. 
Crystal growth takes place and a dense uniform mass of ferrite is 
produced. The material is hard and, to avoid excessive working, it is 
often prepared in moulds to give roughly the shape finally required. 

Since the metal atoms in the material are in strong chemical combina- 
tion in the ferrite molecule, the electrons are more tightly bound than 
in pure metals and the resistance of the ferrites is much greater than that 
of the magnetic metals. Typically, the resistivity of a ferrite is many 
million times that of iron. 

The crystal structure of the ferrite consists of a face-centred cubic 
structure of oxygen ions with the metal ions M and Fe fitted in between. 






124 



LOW NOISE ELECTRONICS 



There are two possible types of site for these metal ions and the two 
ferric ions lie on different types of site. Thus, for example, in a nickel 
ferrite where M is nickel, one ferric ion and the nickel ion are on one 
type of site, and the other ferric ion is on the other type of site. The 
siting of the metal ions is important in establishing the difference 
between the origins of the magnetism in the pure metals (ferromag- 
netism) and in the ferrites (ferri magnet ism). 

In the ferromagnetic metals the magnetic moments of all the atoms in 
any one region or domain are aligned in the same direction. In the un- 
magnetised state the magnetic moments of adjacent domains are 
orientated at random, so that there is no nett magnetisation. An 
applied steady magnetic field causes whole domains to swing round so 
that their magnetic moments lie in the direction of the field. 

In the ferrimagnetic material the magnetic moments of some of the 
atoms in the molecule oppose each other. In particular, in nickel 
ferrite the magnetic moments of the atoms on different types of site 
tend to oppose each other. Thus the resultant magnetic moment per 
molecule is not that due to the arithmetic sum of two ferric atoms and a 
nickel atom, but is much closer to that of the nickel atom alone. It is 
the resultant magnetic moments of the ferrite molecules which are 
aligned in the same direction in any one domain. 

Gyromagnetic Effects 

A spinning electron possesses a magnetic moment and an angular 
momentum which are in opposite directions. If a steady magnetic 
field Ho is applied to a magnetic material the direction of the magnetic 
moment of the spinning electron tends to align itself with the field. 
However, due to the gyroscopic effect of the associated angular mo- 
mentum, precession takes place and the end of the magnetic moment 
vector describes a circle around the direction of the field Bk in a plane 
perpendicular to that direction. Due to damping forces in the material 
the amplitude of the precession gradually decreases, i.e. the circle gets 
smaller, until the magnetic moment of the spinning electron and the 
direction of the field are aligned. The material is then magnetised. The 
frequency of the precession depends on the strength of Ho. 

If the material is placed in a steady field Ho so that the spin magnetic 
moments are already aligned, then the application of an alternating 
magnetic field with a component perpendicular to Ho will disturb this 
alignment and precession will occur. Interaction takes place between 
the precessing magnetic moments of the spinning electrons and the 
alternating magnetic field. This interaction produces phase changes, 
attenuation, etc., in the alternating field, and these effects are used in the 



FERRITES AND THEIR MICROWAVE APPLICATIONS 



125 



microwave devices described below. The interaction is dependent on 
the frequency of the alternating field for a given fit, and upon the 
polarisation of the alternating field relative to the direction of Ho. 

The state of polarisation of the alternating signal applied to the 
ferrite can invariably be described in terms of circular polarisation. 
Thus the commonly encountered plane polarised wave is equivalent 
to two circularly polarised waves of equal amplitude but of opposite 
senses of polarisation, i.e. one clockwise and one anti-clockwise. 

In a magnetised ferrite there are two different phase velocities for the 
two different components of the applied alternating signal. That com- 
ponent of the signal which is circularly polarised in a clockwise sense 
about the direction of the steady field has one phase constant (p+), and 
the component which is circularly polarised in the opposite sense is 
propagated with a different phase constant ((3-) in any direction in the 
material. 

After passing through a thickness d of magnetised ferrite two waves 
circularly polarised in opposite senses about the steady field will each 
undergo different phase changes and, if originally in phase, will emerge 
with a phase difference (p+ — $-)d between them. 

In addition to this differential phase change for the two senses of 
circular polarisation, there may be differential absorption. 

For a given steady field Ho there is a certain precession frequency for 
the spinning electron. If an alternating signal of this frequency and 
circularly polarised in a positive sense is applied to the ferrite, then 
resonance occurs and a large amplitude of precession is excited. There 
is consequently a large absorption of energy from the signal due to the 
forces damping the precession. Had the signal been polarised in the 
opposite sense, then this resonance absorption would not have occurred. 

Thus the differential behaviour of magnetised ferrites is dependent 
upon the sense of the circular polarisation of the applied microwave 
signal relative to the steady field. There are two such differential 
phenomena: in the first case the phase constant of the positively circular 
polarised wave differs from that of the negatively polarised wave; while 
in the second case resonance absorption occurs for the positively 
polarised wave but not for negative polarisation. 

For ferromagnetic resonance at microwave frequencies high values of 
Ho are needed, and devices depending on resonance absorption are 
sometimes called strong field devices. If the other differential effect 
depending on phase constant is to be employed, then low values of Ho 
are used so that there is no possibility of resonance absorption intro- 
ducing unwanted effects. 

An appreciation of gyromagnetic behaviour and its applications is 






126 



^ 



LOW NOISE ELECTRONICS 



probably best obtained by considering the individual microwave 
devices described below. 

Phase Shifter for Circularly Polarised Waves 

A ferrite rod is mounted in polyfoam along the axis of a circular wave- 
guide. A magnetic field H Q is directed along the length of the rod by a 
solenoid carrying current. If a wave of positive circular polarisation is 
passed along the guide, then it will undergo a phase change which will 
depend upon the length of ferrite traversed and upon the field // . The 
phase change can be varied by varying the current through the solenoid. 
A different phase change would be obtained if the same wave were sent 
through the guide in the opposite direction because it would now be 
negatively circularly polarised about the field direction. 

Rectangular waveguide systems carrying plane polarised waves are in 
practice more common than circular guides. The variable phase shifter 
described above can, however, be used in a rectangular system by in- 
serting two quarter wave plates mutually perpendicular, one before and 
one after the ferrite rod. 

A quarter wave plate may consist of an insulating plate, or a set of 
vanes, to introduce a phase change of 90° between components per- 
pendicular and parallel to the plate. Thus a plane polarised wave is 
converted to a circularly polarised wave if the quarter wave plate is set 
at 45° to the plane of polarisation of the plane polarised wave. 

The first quarter wave plate converts the plane polarised wave to 
circular polarisation so that it may be operated on by the ferrite rod. 
The circularly polarised wave, delayed in phase by a given amount after 
passing through the rod, is now reconverted to a plane polarised wave 
by the second quarter wave plate. 

Faraday Rotation — Isolator — Gyrator 

It can be shown that a plane polarised wave is equivalent to two 
circularly polarised waves of equal amplitude but opposite sense. These 
two circularly polarised waves will be propagated through a longi- 
tudinally magnetised ferrite with different phase constants, so that when 
they are compounded after passing through the ferrite, the direction of 
plane polarisation of the resultant is different from that incident on the 
rod. The plane of polarisation has been rotated by an amount depend- 
ing upon the longitudinal magnetic field and the length of ferrite. This 
phenomenon is known as Faraday rotation, and in conjunction with 
absorbing elements set at the correct angle, or suitably twisted wave- 
guide, it is used as the basis of isolators or gyrators. 
A resistive card in a waveguide will strongly attenuate a wave passing 









FERRITES AND THEIR MICROWAVE APPLICATIONS 



127 






over it with its electric vector polarised in the plane of the card, while a 
wave polarised perpendicularly to the card will hardly be affected. 

Two such cards are placed, one at each end of a longitudinally 
magnetised ferrite rod mounted in a circular guide so that a 45° clock- 
wise Faraday rotation is obtained for signals going from left to right. 
The resistive card on the right is at 45° to the other card, also in the 
clockwise direction viewed from the left. A signal entering the system 
from the left and passing unattenuated over the left-hand card is still 
polarised perpendicular to the plane of the right-hand card and is un- 
attenuated. 

A similar signal entering the system from the right and passing un- 
attenuated over the right-hand card will again be rotated through 45° 
in the same direction as before — clockwise about the direction of the 
field— so that now it meets the left-hand card polarised in the plane of 
the card and is highly attenuated. 

Circular-to-rectangular tapers are provided to allow the isolator to 
be used in a normal rectangular waveguide system. The rectangular 
tapers at each end of the isolator described above will be twisted 45° 
relative to each other just like the resistive cards. 

In the gyrator a 90° twist is put in the taper on one side of a circular 
guide containing a ferrite rod giving 90° clockwise rotation about a 
steady magnetic field directed from left to right. For a signal going 
from left to right, say, the 90° twist is added to the 90° rotation giving 
180° rotation, which is equivalent to a phase change of 180°. For a 
signal going the opposite way the 90° twist is subtracted from the 90° 
rotation giving no resultant effect. 

Circulator 

A four-port circulator based on Faraday rotation is produced by having 
four rectangular guide inserts into the circular guide. The guide con- 
tains a magnetised ferrite rod which gives a 45° rotation. The two in- 
serts on the left of the rod (ports 1 and 3) are at right angles to each 
other, so that no signal will pass between them. The two inserts on the 
right of the rod (ports 2 and 4) are also mutually perpendicular and 
twisted 45° relative to 1 and 3. The senses of the Faraday rotation and 
the twist are such that there is no nett rotation in going from 1 to 2, but 
90° in going from 2 to 1 and hence no coupling. Similarly, signal will 
pass from 2 to 3, 3 to 4, and 4 to 1 but not in the opposite sense. 

Transverse Fields — Rectangular Guides 

The devices so far described all depend upon a longitudinally magnetised 
ferrite rod in a circular waveguide which is fitted into a standard 



128 



LOW NOISE ELECTRONICS 



B 



© 



rectangular system by suitable tapers. It is, however, possible to get 
gyrator and isolator effects directly in rectangular guides by putting a 
ferrite slab near and parallel to one of the short sides of the rectangular 
section, with Ho applied perpendicular to the long 
sides. 

The instantaneous magnetic field in a rect- 
angular guide is shown in Fig. IV.2, the loops 
moving in the direction of propagation. Let us 
assume that the wave is propagated up the page. 
Then a point on the side A will experience a 
magnetic field vector rotation anticlockwise in the 
plane of the paper, while on the side B the vector 
— < — . will rotate in the opposite sense about the steady 

transverse field H Q directed into the paper. If the 
Fig, IV.2 direction of propagation is now reversed, then the 

sense of the rotation at A and B will also be 
reversed. Thus if a ferrite slab is placed near the side A, then there will 
be different phase velocities for waves propagated in different directions 
and hence a unidirectional nett phase shift. 

If the field Ho is increased until ferromagnetic resonance occurs, then 
the unidirectional gyrator described above becomes an isolator. 

Other Ferrite Devices 

The microwave devices discussed briefly above represent only a few of 
the applications of the special properties of ferrites. Other properties 
associated with gyromagnetism, like the concentration of negatively 
polarised energy in the ferrite, are used to give unidirectional micro- 
wave effects. The low frequency uses as cores have been mentioned in 
the introduction, and there is a whole important field of application in 
computing based on the square hysteresis loop which can be obtained 
in certain ferrites. 
The references quoted below give details of most applications. 1 - 2 

1 High Frequency Applications of Ferrites, Roberts, E.U.P. 

2 Proc. I.E.E., Vol. 104, Part B, Supplements 5, 6, 7. 









Appendix V 

The Injection Laser 



laser action in a p-n junction was first obtained in November 1962, 
using gallium arsenide. 

The importance of the injection laser lies in its small size and sim- 
plicity and the fact that it can be modulated directly by varying the 
stimulating current flowing across the p-n junction. In other lasers the 
output is modulated by fairly complicated electro-optical devices 
operating on the light beam itself. The first successful transmission of 
voice signals over an injection laser beam was made in the early summer 
of 1963. 

Two Level Maser Action 

In any system which shows two level maser action the individual units 
of the system have two permitted energy levels £2 and £1, where 



Fig. V.l 

£2 > £1. In a normal system in equilibrium at a given temperature the 
population in level £2 is much smaller than that in £1. 

-<g a -igi) 
N2 — Me kT 

The system is intended to amplify signals of frequency /, where 
£2 — £1 = hf. 

A photon of such radiation entering the system may be absorbed by a 
unit in state £1, which will thereby be excited to state £ 2 . Alternatively 
the incident photon may stimulate one of the units in state £2 to fall to 



130 



LOW NOISE ELECTRONICS 



state E\, emitting as it does so a photon of the same frequency as the 
incident photon and coherent with it. 

There may also be an occasional transition of a unit from E% to £1 
unconnected with the arrival of an incident photon. Such spontaneous 
emission, which is not coherent with the signal, constitutes noise and 
will not be considered further here. 

A photon of energy in the signal incident on the system may thus be 
absorbed if it meets a unit in state £1 or amplified if it meets a unit in 
state E 2 . But since the number (JV2) of units in state £2 is much less than 
the number (M) in state £1 there is a much greater probability of 
absorption than of stimulated emission. A signal incident on a normal 
system in temperature equilibrium with its surroundings will be 
attenuated not amplified. 

If amplification is to be achieved in the material, then a non-equili- 
brium population distribution must be achieved and maintained, i.e. 
Nz must be increased relative to N\. This non-equilibrium state is 
usually obtained by pumping, i.e. by supplying energy from a power 
source to pump units from state £1 to state £2. Thus Ni is decreased 
and JV2 is increased, and the signal power is amplified at the expense of 
power from the pump. 

The required increase in Afe relative to N\ could in principle have been 
achieved just as well not by pumping some of the N\ population up to 
JVa, but by introducing into the system from outside enough extra units 
already in the higher energy state £ 2 . This process is used in the in- 
jection laser. 

Simple Injection Mechanism 

Suppose that a material can exist in the n and p forms due to the 
presence of impurity centres giving the levels shown in Fig. V.2 (i). 

If an abrupt discontinuity between the n and p regions can be 
arranged in such a material, then a rectifying junction will be formed 
(p. 26). 

Such a junction biased in the forward direction by an external battery 
is shown in (ii) and we shall concentrate on that portion of the forward 
current which consists of electrons injected into the p region from the 
n region. This will be the predominant part of the current if the con- 
ductivity of the n-type material is made much greater than that of the 
p-type. The most energetic of the electrons in the conduction band of 
the rt-type material, those shown shaded on the distribution curve, will 
have sufficient energy to pass into the p-type material. 

The p-type material thus has electrons injected into it at a high energy 
level. In terms of the two level discussion above, the population JV2 at 






THE INJECTION LASER 



131 



the higher level £2 is increased relative to the low level population N\. 

A photon of frequency,/ = 2 ~ 1 > travelling in the p-type material 

will thus have an increased chance of stimulating emission of another, 
coherent, photon by causing an electron to faU from £2 to £1. Note 
that it is possible for electrons to fall into £1 because the material is 
p-type with holes in the valence band. 




ELECTRON 

DISTRIBUTION 

_L_ 



Jn 



7[ 



HOLE 

DISTRIBUTION 



(i) 



(ii) 



Fig. V.2 



The photon travelling in the p-type material might have been ab- 
sorbed in exciting an electron from £1 to £2, but the chance of such 
absorption occurring will decrease relative to the chance of stimulated 
emission as the number of electrons injected into £2 across the junction 
increases, i.e. as Nz increases relative to JVi. 

If the forward current is sufficiently increased there will be a nett 
gain of light energy in the p-type material. If two silvered optical flats 
are ground on to the light emitting layer of the diode then the coherent 
photons which are perpendicular to the faces are reflected to and fro 
and are amplified by the mechanism described above. If one of the 
faces is not made fully reflecting, then sufficient energy is fed back to 
maintain the optical resonator in oscillation, while the remainder is 
emitted as a coherent beam. 

Practical Injection Junction Materials 

A population inversion of the type described above in a p-type material 
could equally well have been produced by injecting holes as high energy 
minority carriers into an K-type semiconductor. It should be re- 
membered that if a hole is given extra energy it goes lower in the energy 
level diagram. Thus the injection of holes from a p-type material across 
the junction into the valence band of an w-type material produces the 
required non-equilibrium distribution, and in general the establishment 
of an excess population of high energy minority carriers in either p- or 



132 



LOW NOISE ELECTRONICS 



/j-type material will produce transitions across the energy gap giving 
photon amplification and the possibility of laser action. 

The energy gap in germanium is 0-72 eV, so that the energy involved 
in a transition is 0*72 x 1-6 x I0 _1B joule. 

Thus hf= 072 x 1-6 x lO" 1 * joule. 



and 



X = 



ch 



0-72 x 1-6 x 10^» 
= 17,200 A.U. 



3 x 1Q8 x 6-6 x IP-* * 
0-72 x 1-6 x 10-w 



-m 



The silicon energy gap of 1-09 eV gives a larger transition energy, but 
the wavelength involved is still very long, and transitions in both 
germanium and silicon result in heating of the material and no light 
output. Gallium arsenide has a gap of 1*4 eV and here a direct transi- 
tion gives a wavelength of about 8000 A.U., so that an infra-red beam 
will be emitted. 

If visible light is required, then a material with a larger transition 
energy must be used. Such a material is gallium phosphide where the 
energy gap is 2-25 eV. 

In a transition process momentum must be preserved as well as 
energy, and with the increase in energy gap direct transitions become 
less probable. A transition takes place via an impurity centre which can 
absorb or emit phonons thus simplifying the conservation of energy and 
momentum conditions. Thus an electron does not cross the gap 
directly but goes via an impurity centre (p. 38). 

Suitable impurity must be introduced into the gallium phosphide, so 
that the transition process can occur with the radiation of visible light. 
Conversely, one of the important requirements in preparing the gallium 
phosphide is that other impurity centres which would give rise to non- 
radiative transitions are reduced to a minimum. 

Different impurities give luminescent centres at different depths in 
the energy gap. Doping gallium phosphide with oxygen and zinc pro- 
duces green light and must correspond to a shallow impurity centre near 
the band edge. Doping with oxygen and silicon gives red light, the 
longer wavelength indicating that the impurity centre is in this case 
deeper; it is in fact thought to be about 0-4 eV from a band edge. 

Silicon carbide, which was the material in which Lossev first observed 
electroluminescence, has a large energy gap, but is difficult to manu- 
facture. In general, the greater the energy gap the more difficult is 
manufacture and attention is directed to solving these problems with 
promising new materials and to discovering more about the desirable 
luminescent centres and the undesired non-radiative centres. 



THE INJECTION LASER 



133 



Injection laser action has been observed in a large number of other 
materials, e.g. gallium arsenide, indium antimonide, and mixed crystals 
of indium gallium arsenide. By choosing a given crystal material, or by 
selecting a certain proportion of elements in a mixed crystal, a wide 
choice of wavelengths is available. In a given material the operating 
wavelength may be varied slightly by an applied magnetic field 1 and 
pressure tuning is also possible. 

New materials, including group 1V-V1 compounds are extending 
laser action into the long-wave infra-red where atmospheric "windows'* 
allow useful transmission, and new pumping and tuning methods are 
likely to be used in commercial and military applications in the near 
future. 2 

Limitations of Injection Lasers 3 

The conversion efficiency (electricity to light) falls as the temperature 
rises and the current required for laser action increases. The junction 
laser can, however, be continuously operated at temperatures up to 
about -77° C. 



CURRENT 



Go As 0'4mm. CUBE. 
POLISHED FACES ABCD. 
EFGH 



JUNCTION 



LIGHT 




CURRENT 



Fig. V.3 



The frequency response is limited by the junction capacity and its 
series resistance, but the upper limit is probably several hundred Mc/s. 
The spectral width of the emitted light may however be several A.U. 
which is much greater than that of the ruby or gas laser, but careful 
control of operating conditions can considerably reduce this. 

1 "Semiconductor Lasers", Quist, International Science and Technology, February 
1964. 

2 "Semiconductor Lasers", Nathan, P.f.EEE., Vol. 54, October 1966. 

3 "Considerations on the Applications of Semiconductor Lasers", Milsum, I.E.E. 
Laser Symposium, September 1964. 



134 



LOW NOISE ELECTRONICS 



A publicised feature of the laser is the extreme narrowness of the light 
beam which is determined only by diffraction from the emitting area. 
The active area of a single junction laser (Fig. V.3) is, however, so small 
that diffraction gives a beamwidth of several degrees rather than the 
fraction of a degree which may be obtained with other types of laser 
beam. 






Appendix VI 

The Requirements for Low Noise 
Amplification Near 1 Gc/s 



the concentration of the new types of low noise amplifying systems 
in radio-astronomy on frequencies in the general vicinity of 1 Gc/s is a 
combination of accident and design. 

A highly directional system must operate in the many megacycles per 
second region if the aerial is to be of reasonable size, and so too must 
any system requiring signals to penetrate the ionosphere. Further- 
more, background noise from the sky falls to a minimum at about 
1 Gc/s. 

Unfortunately the excess noise introduced by U.H.F. triode and 
transistor amplifiers increases very sharply between 100 Mc/s and 
1 Gc/s, while crystal mixers also have high noise figures in this fre- 
quency range. Thus any conventional amplifier introduces noise which 
is several orders of magnitude greater than the natural background. 

Typically at 1 Gc/s the noise temperature of a conventional radio 
receiving system might be about 2000° K, while the sky temperature is 
only about 10° K. If the system noise could be reduced to about the 
same order as the sky noise, then the sensitivity would be enormously 
increased. 

Such an improvement is possible if a parametric amplifier or a maser 
is used in the receiver. Parametric amplifiers have a high frequency 
limit of several Gc/s and masers a low frequency limit of about I Gc/s. 
Between them these devices cover the whole frequency range where the 
sky noise is low (about 0-5-10 Gc/s). In particular, either can be used 
at 1420 Mc/s, the hydrogen line frequency (see p. 138), which is of vital 
interest in radio astronomy. 

System Noise 

System noise is made up of the following components: aerial overspill, 
aerial and feeder losses, device noise, and second stage noise. 



136 



LOW NOISE ELECTRONICS 



Aerial Overspill Back lobes and side lobes of the directional aerial 
system may be pointed at noise sources, e.g. the ground, at relatively 
high temperatures like 300° K. 

Aerial and Feeder Losses. Any attenuating element introduces into 
the system an amount of noise dependent on the attenuation and the 
temperature of the aerial, switch, waveguide, etc., producing the 
attenuation (see p. 9). 

Device Noise. The noise introduced by whatever device is used for 
the first stage of amplification is the principal component of the noise 
produced in the receiver itself. Typical noise temperatures of some 
conventional devices used at these frequencies are: 

Triode valve 120" K at 100 Mc/s, 1200° K at 1 Gc/s. 
Crystal mixer 500-1000° K. 
Transistor 1500-2000° K. 
Tunnel diode amplifier 1000° K. 

Second Stage Noise. In a low noise system the noise introduced by 
the second stage may be significant. Thus a conventional second 
stage with a noise temperature of 1000° K, following a first stage 
which has a power gain of 100, will add 10° K to the system noise 
temperature. 

AH these factors must be considered if a system is to be produced with 
the lowest noise temperature. The substitution of a low noise amplifier 
for the conventional first stage will make it worth while to redesign the 
associated aerial, feeders, etc. for low noise properties. 

Three stages in the improvement of the total noise temperature of a 
system might be: 

Conventional amplifier with conventional aerial, feeders, etc., 
2000° K. 

Low noise amplifier with conventional aerial, feeders, etc., 100- 
200° K. 

High quality specially designed maser system (see p. 9), 18-5° K. 

Practical limit probably a few degrees absolute. 

Sky Noise 

The two components of sky noise are galactic noise and atmospheric 
noise. 

The origin *of galactic noise is uncertain, but it consists of a con- 
tinuous background whose intensity varies with celestial direction but 
not with time. Superimposed on this background are various discrete 



REQUIREMENTS FOR LOW NOISE AMPLIFICATION NEAR 1 Gc/s 137 

sources or radio stars. The background noise falls sharply with fre- 
quency, in the manner shown in Fig. VI. 1, so that the noise temperature 
is about 100° K at 100 Mc/s but only a few degrees absolute at 1 Gc/s. 



MASERS »- 



-* PARAMETRIC AMPLIFIERS 



1000 



100 




10 10 

FREQUENCY IN Gc/s (10 3 c/s) 



100 



RADIO TELESCOPES JANUARY 1963 



O PARAMETRIC AMPLIFIER 
• MASER 



IN EACH CASE THE TOTAL SYSTEM 
OF INSTALLATION IS PLOTTED. 
BRITISH STATIONS LABELLED. 



NOISE 






NOTE:- ALL PARAMETRIC SYSTEMS ARE ABOVE 100°K. 

ALL MASERS BELOW, ALTHOUGH OVERLAP HAS NOW OCCURRED 

Fig. VI. 1 



Energy is absorbed when it passes through the Earth's atmosphere 
and the attenuation results in noise as shown in Chapter I, p. 9. The 
mechanism by which oxygen and water molecules absorb energy has 
been discussed in Chapter 2, p. 14. 

Figure VI. 1 shows the way in which atmospheric noise increases with 



138 



LOW NOISE ELECTRONICS 



frequency, and the position of the noise peaks. The minima or 
"windows" above 100 Gc/s are of interest in millimetric systems, but 
the attenuation in this region due to water vapour appears higher than 
expected, particularly near oxygen lines. 1 

Figure VI. 1 also shows galactic noise becoming negligible at fre- 
quencies above a few Gc/s, and atmospheric noise becoming negligible 
at frequencies below 1 Gc/s. Thus the total sky temperature has a 
minimum value at a few Gc/s, and a generally low value between, say, 
0-5 and 10 Gc/s. The indicated operating bands of masers and para- 
metric amplifiers both extend into this low sky temperature region and 
the two bands overlap. Either device may thus be used at 1420 Mc/s, 
the frequency of the hyperfine hydrogen line (see below). 

Also shown on the diagram (dots) are the noise temperatures of 
current low noise systems (see below) and of conventional triodes and 
crystal mixers. 

Hydrogen 1420 Mc/s Line 

Hyperfine structure in spectra is caused by nuclear spin. With the 
nucleus of the hydrogen atom spinning in a given direction there are 
two possible orientations for the spinning electron, each corresponding 
to a given atomic state and energy. Transitions from one of these states 
to the other involve the emission or absorption of a quantum of energy. 
The frequency of this radiation is 1420 Mc/s (compare caesium where 
the single outermost electron reacts with the magnetic moment of the 
spinning nucleus to give the 9192 Mc/s of the caesium atomic clock). 

The probability of a hydrogen 1420 Mc/s transition occurring is very 
low — about one transition per atom in ten million years — but the 
amount of atomic hydrogen in space is so great that it was correctly 
predicted in 1944 that the 1420 Mc/s radiation would be detectable. 

About 90% of the interstellar gas is atomic hydrogen, with most of 
the remainder being ionised hydrogen in the vicinity of the hot stars. 
Radiation from the atomic hydrogen is able to penetrate the dust which 
obscures the centre of the galaxy when viewed with the great optical 
telescopes. 

The concentration of gas at the galactic centre and the spiral form of 
the arms can be observed, and the motion of the gas has been established 
by Doppler frequency shift measurements. 

The existence of the 1420 Mc/s hydrogen line and the fact that it 
occurs in the sky temperature minimum has been of tremendous im- 
portance to the astronomers and cosmologists. 2 

1 Tolbert and Straiton, Proc. I.R.E., March 1961, p. 649. 

2 The Exploration of Outer Space, Lovell, O.U.P. 



. 



REQUIREMENTS FOR LOW NOISE AMPLIFICATION NEAR 1 Gc/s 139 

Existing Low Noise Systems 

The dots in Fig. VI. 1 indicate the total noise temperature of some 
existing low noise installations in the radio astronomy field. Some of 
these systems employ pencil beams through which the signal source 
drifts due to the Earth's rotation, some track the source the whole of the 
time it is above the horizon. Other systems use interferometer tech- 
niques, with two spaced aerials in the simplest case and with multiple 
aerials and elaborate phase switching in the more complex high resolu- 
tion instruments. 3 - 4 

These interferometer techniques distinguish between small and 
extended sources and thus tend to reduce the eifect of the galactic noise 
background. Thus even at the lower frequencies where galactic noise is 
high it is worth using a low noise amplifier. Parametric amplifiers are 
used in this low frequency application. 

In any case the observation or "integration" time needed to detect a 
given signal increases with the system noise and this is an important 
consideration in radio astronomy where a long integration time may 
entail tracking the source with a heavy aerial system. 

In general, the maser systems operate at higher frequencies and have 
lower noise figures than the parametric systems, although they do over- 
lap in frequency and occasionally in noise figure. The parametric 
amplifier, usually employing a solid state diode or an electron beam, is 
a simpler device to operate than the maser. In particular, it does not 
need helium cooling or a magnet, so that it is smaller and lighter to fit 
on the aerial system. The operational balance may be restored by the 
development of compact masers with closed cycle helium liquefiers and 
superconducting magnets. 

Outside the "Window" 

The requirement for low noise amplification arises in earth-bound 
receivers studying radiation from space because of the radio "window" 
at about 1 Gc/s. If, however, the receiver is taken outside the atmo- 
sphere in a satellite or even up in a high flying balloon, then higher fre- 
quencies into the millimetre and infra-red regions can be detected. At 
these very high frequencies compact masers or maser-type photon 
counters are likely to be increasingly employed. 

3 Ryle, Proc. Roy. Soc, Vol. 211 A, 1952, p. 351. 

4 Christiansen and Mathewson, Proc. I.R.E., Vol. 46, 1958, p. 127. 



Bibliography 



General Books 

Wave Mechanics of Crystalline Solids, Smith, Chapman & Hall. 
Physical Electronics, Hemenway et al., Wiley. 

Microwave Tubes and Semiconductor Devices, Sims and Stephenson, 
Blackie. 

Specialised Books 

Coupled Mode and Parametric Electronics, Louisell, Wiley. 
Var actor Applications, Penfield and Rafuse, M.T.T. Press. 
Power Travelling-wave Tabes, Gittins, E.U.P. 
Lasers, Lengyel, Wiley. 

Applied Optics. "Supplement on Optical Masers", 1962, 
P.IE.E.E.j Applied Optics. "Joint Issue on Optical Electronics", 
October 1966. 

Recommended Journals 

International Science and Technology, New York. 

Proceedings of Institute of Electrical and Electronic Engineers; also 
associated publications, I.E.E.E. Spectrum and I.E.E.E. Trans- 
actions on Electronic Devices. 

Journal of Institution of Electrical Engineers. 












Index 



Index 



Abrupt junction, 71 
Absorption coefficient, 8 

— expts. in semiconductors, 39 

— lines, microwave, 16 

— resonance in ferrites, 126 
Absorptive power, 2 
Acceptor level, 24 

Active material, 55 
Adler tube, 81,86 
Aerial losses, 9, 1 36 

— noise input, 6 

— overspill, 1 36 
Afterglow, 36 

Ammonia inversion resonance, 14 

— microwave spectrum, 14 
Amplitron, 120 
Applegate diagram, 109 
Application of lasers, 103 
Arsenic, 24 

Atmospheric absorption, 14 
Atomic clock, caesium, 13 
Atoms complex, spectrum, 12 

Backward wave oscillation, 118 

— tube, 117 

Band system, conductor, 21, 22 

— insulator, 21 

— of solids, 19 
Black body, 1 
Bloembergen, 52, 56 

Bohr theory of hydrogen atom, 1 1 
Boltzmann's constant, 3 

— law, 3 

Brewster windows, 100, 102 
Broadening of spectral line, 15 
Buncher, klystron, 112 
Bunching distance, 113 

Cadmium-copper alloy, 33 
Caesium clock, 13 



Calcium fluoride, 58 
Carcinotron, 121 
Catcher, klystron, 1 12 
Cerium, 54 
Circulator, 68, 123, 127 

— helium cooled, 94 
Coherent source, 59 

Collision broadening, spectral line, 

15 
Colour temperature, 7 
Communication laser, 105 
Complex atoms, spectra, 1 2 
Compton effect, 41 
Conduction, band system, 21, 22 
Conductivity, electrical, 32 
Conductivity, thermal, 34 
Conductor, 22 
Confocal, laser mirror, 100 
Constant temperature enclosure, 1 
Continuous spectrum radiation, 1 
Conversion efficiency, 133 
Copper-cadmium alloy, 33 
Cross-field tubes, 120 
Crystal momentum, 37 
Cuccia coupler, 81 
Cyclotron, 82 

— wave, 81 

— wave, fast and slow, 82 
CW laser, 101 

de Broglie, 41 
Degenerate amplifier, 68 
Depletion layer, 27, 71 
Diffusion, electrons and holes, 27 
Diode, Esaki, 28 

— p-n junction, 26 

— tunnel, 28 

— varactor, 70 

— variable capacity, 70 
Direct gap semiconductor, 37 






146 

Discrete frequency spectrum, 1 1 

Donor level, 24 

Doping, 23 

Doppler broadening, spectral line, 1 5 

— effect, Mdssbauer, 44 

— shift, 138 
Down converter, 66 
Duplexer, 92 

Effective mass of electron, 29 
Einstein radiation theory, 44, 51 
Elastance, 74 
Elastic waves, 33 
Electrical resistance, 34 
Electron beam amplifiers, 108 

— beam paramp., 81 

— beam tubes, practical, 1 19 

— coupler, 83 

— gun, low noise, 87 
photon interaction, 35 

— trap, 36 
Emission induced, 47 

— spontaneous, 51 

— stimulated, 50 
Emissive power, 2 

Enclosure, constant temperature, 1 
Energy bands, 19 

— level splitting, 19 
Equaliser, 92, 93 
Exciton, 38 

Exclusion, Pauli principle, 1 8 
External energy processes, 1 

Fabry-Perot, 56 

Faraday rotation, 126 

Fast and slow plasma waves, 1 1 1 

Fast cyclotron wave, 82 

Feeder loss, 9 

Ferrimagnetism, 124 

Ferrites, 122 

Ferrite crystal structure, 1 23 

— in transverse field, 127 

— maser switch, 93 

— paramp., 70, 79 

— precession in, 1 24 

— resistivity, 123 

— sintering, 123 
Ferro-electric paramp., 80 
Filter, bridged tee, 93 
Fine structure spectrum, 12 



INDEX 



Gadolinium, 54 
Gain-bandwidth, maser, 57 
Galactic noise, 137 
Gallium arsenide, 29, 37 

— arsenide energy gap, 132 

— phosphide, 24 

— phosphide energy gap, 1 32 
Gas laser, 58 

Germanium, 26 

— energy gap, 132 
Graded junction, 71 
Group velocity, 1 1 1 

Gun electron, low noise, 87 
Gunn effect, 28 
Gyrator, 123, 126 
Gyromagnetic effects, 124 

Harmonic generator, 40 

Heating effects in laser, 99 

Helium-neon laser, 58, 101 

Helix, 114 

Hole, 24 

Hot electrons, 29 

Hydrogen atom, Bohr theory, 1 1 

— atom, energy levels, 17 

— line, 13, 138 
Hydroxy!, microwave line, 16 
Hyperfine structure spectrum, 1 2 

Idler frequency, 61 
Impedance matrix, 73 
Impurity centre, 36 

— level, 23 

— semiconductor, 23 
Indirect gap, 38 
Indium, 24 

— antimonide, 133 

— gallium arsenide, 133 
Induced absorption coefficient, 45 

— emission, 47, 50 

— emission coefficient, 45 
Injection junction materials, 131 

— laser, 129 

— laser array, 1 02 
Insulator, 22 

— band system, 21 
Integration time, 139 
Interferometer techniques, 1 39 
Internal energy processes, 31 
Interstellar gas, 13, 16, 138 
Intrinsic semiconductor, 25 



INDEX 



147 



Inversion, population, 51 
Inversion resonance, ammonia, 14 
lonisation energy, 1 7 
Iron, radioactive, 43 
Isolator, 123, 126 

Junction, abrupt, 71 

— graded, 71 

— laser, 129 

— p-n, capacity, 70 

Kerr cell, 101 

Kirchhoff's radiation law, 2 
Klystron amplifier, 112 
KTB, 5 

Lanthanum ethyl sulphate, 54 
Laser applications, 103 

— communication and ranging, 
105 

— conversion efficiency, 133 

— gas, 58 

— heating effects, 56, 99 

— injection, 129 

— injection array, 102 

— solid, heating effects, 56 

— systems, 98 

— welding and cutting, 104 
Lattice energy, 32 

Line spectrum, 1 1 
Lithium energy levels, 18 
Lobes subsidiary, noise, 9 
Lossev, 132 

Lossy elements, noise, 94 
Lossy medium noise, 8 
Lower sideband paramp., 61 

Magnetite, 123 

Magnet, maser, 92 

Magnetron, 120 

Manley-Rowe, 61 

Manley-Rowe general relations, 69 

Mars radar experiment, 91 

Maser, cavity, 90 

— future, 95 

— gain-bandwidth, 92 

— magnet, 92 

— noise temperature, 8, 92 

— optical pump, 96 

— peak pulse in radar, 93 

— phonon, 59 



Maser, practical system, 90 

— stagger tuned, 92 

— Telstar, 91 

— travelling wave, 91 
Mass effective, electron, 29 
Matrix, varactor impedance, 73 
Microwave spectroscopy, 13 
Mirrors, confocal laser, 100 
Mobility, 32 

Modes of optical resonator, 99 
Modes per unit volume, 47 
Modulation, laser, 106 
Molecular structure, 13 
Momentum of radiation, 41 
Mdssbauer effect, 43 
M-type tubes, 120 
Multiquantum effects, 105 

Natural width of spectral line, 15 
Negative resistance, 28 
Neutralised plasma, 109, 110 
Noise, reduction by cooling, 94 

— due to lossy medium, 8 

— electronic, 4 

— feeder loss, 9 

— figure, 7 

— input to aerial, 6 

— input temperature, 7 

— laser, 51 

— maser and laser, 49 

— mixer, 135, 136 

— radome, 96 

— reduction in electron beam 
tubes, 118 

— second stage, 136 

— sky, 9, 136 

— stripping, 89, 118 

— subsidiary lobe, 9 

— system, 135 

— temperature, 7 

— temperature, complete system, 9 

— temperature, maser, 92 

— temperature, radar, 94 

— transistor, 135, 136 

— triode, 135, 136 

— window, 133, 138 
Non-equilibrium systems, 50 
Non-linear photon processes, 40 
Non-linear quantum effects, 105 
H-type semiconductor, 23 
Nuclear recoil, 43 



148 

Nyquist, 5 

One-port device, 68 

Optical pumping, mascr, 57, 96 

O-typc tubes, 1 20 

Oxygen, molecular spectrum, 14 

Parametric amplifier (paramp.) 

— Adler tube, 86 

— D.C pumped, 89 

— degenerate, 68 

— diode, 70 

— diode practical circuit, 77 

— electron beam, 81 

— equivalent circuit, 67 

— ferrite, 70, 79 

— ferroelectric, 80 

— performance, 79 

— practical circuits, 67 

— quadrupolc, 86 

— straight, 64 

— travelling wave, 79 

— up, down converter, 66 
Pauli exclusion principle, 18 
Phase constant, 1 1 1 

Phase velocity, 111 
Phonon, 32, 53, 57, 59 
Phosphor, 36 

Photon frequency doubling, 40 
Planck distribution law, 2, 4 
Planck radiation expression, 47 
Plasma, 109 

— frequency, 110 

— neutralised, 109 

— oscillations, 110 

— reduced frequency, 1 10 

— wave, fast and slow, 1 1 1 
p-u junction, 26 
Poisson's equation, 72 
Population inversion, 51 
Potential hill, 27 

Power relations in non-equilibrium 

systems, 60 
Pressure broadening, spectral line, 15 
i?-type semiconductor, 23 
Pulse operation, laser, 56 
Pumping, maser, 52 

— methods, 56 

— optical, 57 

— saturation, 53 
Pump, klystron, 92 



INDEX 

Q spoiling, 101 
Quadrupole, 84 
Quadrupole amplifier, 86 
Quantum, non-linear effects, 105 
Quarter-wave plate, 126 

Radiation, continuous spectrum, 1 
Radio stars, 137 
Radio telescopes, 137 
Radome noise, 96 
Raman effect, 105 
Rayleigh-Jeans law, 4 
Receiver noise figure, 7 
Recoil, nuclear, 43 
Recombination centre, 36 
Rectification p-n junction, 27 
Reduced plasma frequency, 1 10 
Refrigeration, maser, 94 
Relaxation time, 32 
Resistance, electrical, 32, 34 
Resonance absorption in ferrites, 125 
Reverse current, 28 
Ruby, 56, 90, 91 

Sapphire trumpet, 100 
Second stage noise, 136 
Selective absorption in semiconduc- 
tors, 39 
Semiconductor impurity, 39 

— intrinsic, 25 

— direct gap, 37 

— indirect gap, 38 

Sideband, upper and lower, paramp., 

61 
Silicon, 24 

— carbide, 1 32 

— energy gap in laser, J 32 
Sky noise, 9, 15, 136 

— temperature, 1 5 

— zenith, 9 

Slow cyclotron wave, 82 
Slow wave structure, 1 14 
Space charge waves, 108 
Spectrum, fine structure, 12 

— hyperfine structure, 12 
Spin-lattice interaction, 53 

— relaxation time, 53 
Spin-spin interaction, 54 

— relaxation time, 54 
Splitting of energy levels, 19 
Spoiled Q, 101 



INDEX 



149 



Spontaneous emission, 51 

— coefficient, 45 

— probability, 15 
Stabilotron, 120 
Stagger tuned maser, 92 
Stark effect, 12 
Stefan's constant, 2 

— law, 2 
Stimulated emission, 50 
Straight amplifier, 64 
Strong field devices, 125 
Superconducting magnet, 139 
Superconduction, 32, 33 
System noise, 135 

Telstar maser, 9 1 , 95 
Temperature, colour, 7 

— noise, 7 

— noise input, 7 
Terminal level, 57 
Thermal conductivity, 34 
Three-level system, 52 

Tool, welding and cutting, laser, 104 
Trap, electron, 36 
Travelling wave maser, 91 

— paramp., 79 

— tube, 1 14 

— tube attenuator, 1 1 6 



Travelling wave, tube stability, 116 
Trumpet, sapphire, 100 
Tunnel diode, 28, 136 

Ultrasonic shutter, 103 
Up converter, 66 
Upper sideband paramp., 61 
Uranium doping, 58 

Varactor diode, cooled, 79 

— harmonic generator, 79 

— practical circuit, 78 
Velocity modulation, 109 

Water, molecular spectrum, 1 4 
Waveguide, corrugated, 1 14 
Waves, fast and slow, 1 1 1 
Weiss derivation of Manley-Rowe, 

62 
Welding, laser, 104 
Wiedermann-Franz law, 34 
Wien radiation law, 44 

Xenon flash tube, 57 
X-rays, 42 

Zceman effect, 1 2 
Zenith, sky noise at, 9 



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